Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types

TL;DR

This paper characterizes and classifies centrally generated primitive ideals of the universal enveloping algebra of the nilradical in complex semisimple Lie algebras, especially focusing on exceptional types, using the Dixmier map and Kostant cascade.

Contribution

It provides an explicit description of centrally generated primitive ideals for exceptional Lie algebras, extending previous results from classical types to exceptional cases.

Findings

Explicit characterization of primitive ideals in exceptional types

Classification of centrally generated primitive ideals for any semisimple algebra

Extension of classical results to exceptional Lie algebras

Abstract

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\mathfrak{b}$ be a Borel subalgebra of $\mathfrak{g}$, $\mathfrak{n}$ be the nilradical of $\mathfrak{b}$, and $U(\mathfrak{n})$ be the universal enveloping algebra of $\mathfrak{n}$. We study primitive ideals of $U(\mathfrak{n})$. Almost all primitive ideals are centrally generated, i.e., are generated by their intersections with the center $Z(\mathfrak{n})$ of $U(\mathfrak{n})$. We present an explicit characterization of the centrally generated primitive ideals of $U(\mathfrak{n})$ in terms of the Dixmier map and the Kostant cascade in the case when $\mathfrak{g}$ is a simple algebra of exceptional type. (For classical simple Lie algebras, a similar characterization was obtained by Ivan Penkov and the first author.) As a corollary, we establish a classification of centrally generated primitive ideals of $U(\mathfrak{n})$ for an arbitrary semisimple algebra $\mathfrak{g}$.

Tables

	$D$	$D^{\prime }$		$D$	$D^{\prime }$		$D$	$D^{\prime }$
1	$010000$	$010000$	2	$011211$, $111221$, $112210$	$111221$, $112210$	3	$011111$, $111110$, $112321$	$111110$, $112321$
4	$011211$, $111210$, $112221$	$011211$, $111210$	5	$111111$, $112321$	$111111$, $112321$	6	$011210$, $111221$, $112211$	$111221$, $112211$
7	$011110$, $111111$, $112321$	$011110$, $111111$, $112321$	8	$010000$, $011210$, $111211$, $112221$	$010000$	9	$001000$, $122321$	$001000$
10	$001100$, $000111$, $122321$, $101110$	$001100$, $000111$	11	$001111$, $122321$, $101110$	$001111$, $101110$	12	$001111$, $000100$, $122321$, $101110$	$001111$, $101110$
13	$001000$, $101111$, $122321$	$001000$	14	$001100$, $000110$, $122321$, $101111$	$001100$, $000110$

	$D$	$D^{\prime }$		$D$	$D^{\prime }$		$D$	$D^{\prime }$
1	$1000$	$1000$	2	$1231$, $1222$	$1231$, $1222$	3	$1121$, $1342$	$1121$, $1342$
4	$1000$, $1221$, $1242$	$1000$	5	$1000$, $1232$	$1000$	6	$1000$, $1220$, $1232$	$1000$
7	$1120$, $1122$, $1342$	$1120$, $1122$, $1342$	8	$1000$, $1220$, $1222$, $1242$	$1000$	9	$0100$, $2342$	$0100$
10	$0111$, $0120$, $2342$	$0111$, $0120$	11	$1121$, $1342$, $2342$	$1121$, $1342$	12	$1100$, $1121$, $1342$, $2342$	$1100$, $1121$, $1342$
13	$0100$, $0122$, $2342$	$0100$	14	$1110$, $0122$, $1342$, $2342$	$1110$, $1342$

$D$	Type of $c$	$D$	Type of $c$	$D$	Type of $c$
1	$(0,0,0,0)$	2	$(0,0,0,c_{{\beta }_4})$	3	$(0,0,c_{{\beta }_3},0)$
4	$(0,0,c_{{\beta }_3},c_{{\beta }_4})$	5	$(0,c_{{\beta }_2},0,0)$	6	$(0,c_{{\beta }_2},0,c_{{\beta }_4})$
7	$(0,c_{{\beta }_2},c_{{\beta }_3},0)$	8	$(0,c_{{\beta }_2},c_{{\beta }_3},c_{{\beta }_4})$	9	$(c_{{\beta }_1},0,0,0)$
10	$(c_{{\beta }_1},0,0,c_{{\beta }_4})$	11	$(c_{{\beta }_1},0,c_{{\beta }_3},0)$	12	$(c_{{\beta }_1},0,c_{{\beta }_3},c_{{\beta }_4})$
13	$(c_{{\beta }_1},c_{{\beta }_2},0,0)$	14	$(c_{{\beta }_1},c_{{\beta }_2},0,c_{{\beta }_4})$

	$D$	$D^{\prime }$	Type of $c$		$D$	$D^{\prime }$	Type of $c$
1	$31$	$31$	$(0,0)$	2	$31,$ $11$	$31,$ $11$	$(0,c_{{\beta }_2})$

	$D$	$D^{\prime }$		$D$	$D^{\prime }$		$D$	$D^{\prime }$
1	$0100000$	$0100000$	2	$0101111$, $1234321$, $1011111$	$0101111$, $1234321$, $1011111$	3	$0101110$, $1112211$, $1234321$, $1011110$	$0101110$, $1112211$, $1234321$, $1011110$
4	$0101111$, $1112210$, $1234321$, $1011111$	$0101111$, $1112210$, $1234321$, $1011111$	5	$1223210$, $1122111$, $1123321$	$1223210$, $1122111$, $1123321$	6	$0001111$, $1223211$, $1122221$, $1123210$	$0001111$, $1223211$, $1122221$, $1123210$
7	$0001110$, $1223321$, $1122110$, $1123211$	$0001110$, $1223321$, $1122110$, $1123211$	8	$0001111$, $1223321$, $1122111$, $1123210$	$0001111$, $1223321$, $1123210$	9	$0101000$, $1111110$, $1112211$, $1234321$	$0101000$, $1111110$, $1112211$, $1234321$
10	$0111111$, $1223210$, $1112211$, $1123221$	$0111111$, $1112211$, $1123221$	11	$0111110$, $1223211$, $1112110$, $1123321$	$0111110$, $1223211$, $1112110$	12	$0111111$, $1223210$, $1112111$, $1123321$	$0111111$, $1112111$
13	$0101000$, $1223321$, $1122110$, $1123211$	$0101000$, $1223321$, $1122110$, $1123211$	14	$0101111$, $1223210$, $1122211$, $1123221$	$1122211$, $1123221$	15	$0101110$, $1223211$, $1122110$, $1123321$	$0101110$, $1223211$, $1122110$
16	$0101000$, $0001111$, $1223321$, $1122111$, $1123210$	$0101000$, $0001111$, $1223321$, $1123210$	17	$1112110$, $1112211$, $1234321$	$1112110$, $1112211$, $1234321$	18	$0000011$, $1112110$, $1112211$, $1234321$	$0000011$, $1112110$, $1112211$, $1234321$
19	$0000110$, $1112110$, $1112211$, $1234321$	$0000110$, $1112110$, $1112211$, $1234321$	20	$0000111$, $1112111$, $1112210$, $1234321$	$1112111$, $1112210$, $1234321$	21	$1224321$, $1122110$, $1122211$	$1122110$, $1122211$
22	$0000011$, $1224321$, $1122110$, $1122211$	$0000011$, $1122110$, $1122211$	23	$0000110$, $1224321$, $1122110$, $1122211$	$0000110$, $1122110$, $1122211$	24	$0000110$, $0000011$, $1224321$, $1122110$, $1122211$	$0000110$, $0000011$, $1122110$, $1122211$
25	$0100000$, $1112110$, $1112211$, $1234321$	$1112110$, $1112211$, $1234321$	26	$0100000$, $0000011$, $1112110$, $1112211$, $1234321$	$0000011$, $1112110$, $1112211$, $1234321$	27	$0100000$, $0000110$, $1112110$, $1112211$, $1234321$	$0000110$, $1112110$, $1112211$, $1234321$
28	$0100000$, $0000111$, $1112111$, $1112210$, $1234321$	$1112111$, $1112210$, $1234321$	29	$0100000$, $1224321$, $1122110$, $1122211$	$1122110$, $1122211$	30	$0100000$, $0000011$, $1224321$, $1122110$, $1122211$	$0000011$, $1122110$, $1122211$
31	$0100000$, $0000110$, $1224321$, $1122110$, $1122211$	$0000110$, $1122110$, $1122211$	32	$0100000$, $0000110$, $0000011$, $1224321$, $1122110$, $1122211$	$0000110$, $0000011$, $1122110$, $1122211$	33	$1223221$, $1123321$	$1223221$, $1123321$
34	$0000001$, $1223221$, $1123321$	$1223221$, $1123321$	35	$0101100$, $1112221$, $1234321$, $1011100$	$0101100$, $1112221$, $1234321$, $1011100$	36	$0101100$, $0000001$, $1112221$, $1234321$, $1011100$	$0101100$, $1112221$, $1234321$, $1011100$
37	$1223221$, $1122100$, $1123321$	$1223221$, $1123321$	38	$0000001$, $1223221$, $1122100$, $1123321$	$1223221$, $1123321$	39	$0001100$, $1223321$, $1122100$, $1123221$	$0001100$, $1223321$, $1123221$
40	$0001100$, $0000001$, $1223321$, $1122100$, $1123221$	$0001100$, $1223321$, $1123221$	41	$0101000$, $1111100$, $1112221$, $1234321$	$0101000$, $1111100$, $1112221$, $1234321$	42	$0101000$, $0000001$, $1111100$, $1112221$, $1234321$	$0101000$, $1111100$, $1112221$, $1234321$
43	$0111100$, $1223221$, $1112100$, $1123321$	$0111100$, $1112100$	44	$0111100$, $0000001$, $1223221$, $1112100$, $1123321$	$0111100$, $1112100$	45	$0101000$, $1223321$, $1122100$, $1123221$	$0101000$, $1223321$, $1123221$
46	$0101000$, $0000001$, $1223321$, $1122100$, $1123221$	$0101000$, $1223321$, $1123221$	47	$0101000$, $0001100$, $1223321$, $1122100$, $1123221$	$0101000$, $0001100$, $1223321$, $1123221$	48	$0101000$, $0001100$, $0000001$, $1223321$, $1122100$, $1123221$	$0101000$, $0001100$, $1223321$, $1123221$
49	$1112100$, $1112221$, $1234321$	$1112100$, $1112221$, $1234321$	50	$0000001$, $1112100$, $1112221$, $1234321$	$1112100$, $1112221$, $1234321$	51	$0000100$, $1112100$, $1112221$, $1234321$	$1112100$, $1112221$, $1234321$
52	$0000100$, $0000001$, $1112100$, $1112221$, $1234321$	$1112100$, $1112221$, $1234321$	53	$1224321$, $1000000$, $1122100$, $1122221$	$1000000$	54	$0000001$, $1224321$, $1000000$, $1122100$, $1122221$	$1000000$
55	$0000100$, $1224321$, $1000000$, $1122100$, $1122221$	$1000000$	56	$0000100$, $0000001$, $1224321$, $1000000$, $1122100$, $1122221$	$1000000$	57	$0100000$, $1112100$, $1112221$, $1234321$	$1112100$, $1112221$, $1234321$
58	$0100000$, $0000001$, $1112100$, $1112221$, $1234321$	$1112100$, $1112221$, $1234321$	59	$0100000$, $0000100$, $1112100$, $1112221$, $1234321$	$1112100$, $1112221$, $1234321$	60	$0100000$, $0000100$, $0000001$, $1112100$, $1112221$, $1234321$	$1112100$, $1112221$, $1234321$
61	$0100000$, $1224321$, $1000000$, $1122100$, $1122221$	$1000000$	62	$0100000$, $0000001$, $1224321$, $1000000$, $1122100$, $1122221$	$1000000$	63	$0100000$, $0000100$, $1224321$, $1000000$, $1122100$, $1122221$	$1000000$
64	$0100000$, $0000100$, $0000001$, $1224321$, $1000000$, $1122100$, $1122221$	$1000000$	65	$0100000$, $2234321$	$0100000$	66	$0101111$, $0011111$, $2234321$	$0101111$, $0011111$
67	$0101110$, $0112211$, $0011110$, $2234321$	$0101110$, $0112211$, $0011110$	68	$0101111$, $0112210$, $0011111$, $2234321$	$0101111$, $0011111$	69	$0111000$, $0112210$, $0011111$, $2234321$	$0111000$, $0112210$, $0011111$
70	$0111000$, $0112211$, $0011110$, $0000011$, $2234321$	$0111000$, $0112211$, $0011110$, $0000011$	71	$0111110$, $0112211$, $0011000$, $0001110$, $2234321$	$0111110$, $0112211$, $0011000$, $0001110$	72	$0111111$, $0112210$, $0011000$, $0001111$, $2234321$	$0111111$, $0011000$, $0001111$
73	$0101000$, $0111110$, $0112211$, $2234321$	$0101000$, $0111110$, $0112211$	74	$0101000$, $0111110$, $0112211$, $0000011$, $2234321$	$0101000$, $0111110$, $0112211$, $0000011$	75	$0101000$, $0111110$, $0112211$, $0001110$, $2234321$	$0101000$, $0111110$, $0112211$, $0001110$
76	$0101000$, $0111111$, $0112210$, $0001111$, $2234321$	$0101000$, $0111111$, $0001111$	77	$0101000$, $0111110$, $0112211$, $0011000$, $2234321$	$0101000$, $0111110$, $0112211$, $0011000$	78	$0101111$, $0111100$, $0112110$, $0011111$, $2234321$	$0111100$, $0112110$
79	$0101110$, $0111000$, $0112211$, $0011110$, $2234321$	$0101110$, $0112211$, $0011110$	80	$0101000$, $0111111$, $0112210$, $0011000$, $0001111$, $2234321$	$0101000$, $0111111$, $0011000$, $0001111$	81	$0112110$, $0112211$, $2234321$	$0112110$, $0112211$
82	$0112110$, $0112211$, $0000011$, $2234321$	$0112110$, $0112211$, $0000011$	83	$0112110$, $0112211$, $0000110$, $2234321$	$0112110$, $0112211$, $0000110$	84	$0112110$, $0112211$, $0000110$, $0000011$, $2234321$	$0112110$, $0112211$, $0000110$, $0000011$
85	$0112110$, $0112211$, $0010000$, $2234321$	$0112110$, $0112211$	86	$0112110$, $0112211$, $0010000$, $0000011$, $2234321$	$0112110$, $0112211$, $0000011$	87	$0112110$, $0112211$, $0010000$, $0000110$, $2234321$	$0112110$, $0112211$, $0000110$
88	$0112110$, $0112211$, $0010000$, $0000110$, $0000011$, $2234321$	$0112110$, $0112211$, $0000110$, $0000011$	89	$0100000$, $0112110$, $0112211$, $2234321$	$0112110$, $0112211$	90	$0100000$, $0112110$, $0112211$, $0000011$, $2234321$	$0112110$, $0112211$, $0000011$
91	$0100000$, $0112110$, $0112211$, $0000110$, $2234321$	$0112110$, $0112211$, $0000110$	92	$0100000$, $0112110$, $0112211$, $0000110$, $0000011$, $2234321$	$0112110$, $0112211$, $0000110$, $0000011$	93	$0100000$, $0112110$, $0112211$, $0010000$, $2234321$	$0112110$, $0112211$
94	$0100000$, $0112110$, $0112211$, $0010000$, $0000011$, $2234321$	$0112110$, $0112211$, $0000011$	95	$0100000$, $0112110$, $0112211$, $0010000$, $0000110$, $2234321$	$0112110$, $0112211$, $0000110$	96	$0100000$, $0112110$, $0112211$, $0010000$, $0000110$, $0000011$, $2234321$	$0112110$, $0112211$, $0000110$, $0000011$
97	$0100000$, $0112221$, $2234321$	$0100000$	98	$0100000$, $0112221$, $0000001$, $2234321$	$0100000$	99	$0101100$, $0112221$, $0011100$, $2234321$	$0101100$, $0011100$
100	$0101100$, $0112221$, $0011100$, $0000001$, $2234321$	$0101100$, $0011100$	101	$0111000$, $0112221$, $0011100$, $2234321$	$0111000$, $0011100$	102	$0111000$, $0112221$, $0011100$, $0000001$, $2234321$	$0111000$, $0011100$
103	$0111100$, $0112221$, $0011000$, $0001100$, $2234321$	$0111100$, $0011000$, $0001100$	104	$0111100$, $0112221$, $0011000$, $0001100$, $0000001$, $2234321$	$0111100$, $0011000$, $0001100$	105	$0101000$, $0111100$, $0112221$, $2234321$	$0101000$, $0111100$
106	$0101000$, $0111100$, $0112221$, $0000001$, $2234321$	$0101000$, $0111100$	107	$0101000$, $0111100$, $0112221$, $0001100$, $2234321$	$0101000$, $0111100$, $0001100$	108	$0101000$, $0111100$, $0112221$, $0001100$, $0000001$, $2234321$	$0101000$, $0111100$, $0001100$
109	$0101000$, $0111100$, $0112221$, $0011000$, $2234321$	$0101000$, $0111100$, $0011000$	110	$0101000$, $0111100$, $0112221$, $0011000$, $0000001$, $2234321$	$0101000$, $0111100$, $0011000$	111	$0101000$, $0111100$, $0112221$, $0011000$, $0001100$, $2234321$	$0101000$, $0111100$, $0011000$, $0001100$
112	$0101000$, $0111100$, $0112221$, $0011000$, $0001100$, $0000001$, $2234321$	$0101000$, $0111100$, $0011000$, $0001100$

	$D$	$D^{\prime }$		$D$	$D^{\prime }$		$D$	$D^{\prime }$
1	$01000000$	$01000000$	2	$23354321$, $12343221$, $11233321$	$23354321$, $12343221$, $11233321$	3	$23354321$, $12344321$, $11232110$, $11232211$	$23354321$, $12344321$, $11232110$, $11232211$
4	$23354321$, $12344321$, $11232100$, $11232221$	$23354321$, $12344321$, $11232100$, $11232221$	5	$23454321$, $12243211$, $11232110$, $11233321$	$23454321$, $12243211$, $11232110$, $11233321$	6	$23454321$, $12243221$, $11232100$, $11233321$	$23454321$, $12243221$, $11232100$, $11233321$
7	$23454321$, $12244321$, $11232110$, $11232211$	$23454321$, $11232110$, $11232211$	8	$23454321$, $12244321$, $11232100$, $11232221$	$23454321$, $11232100$, $11232221$	9	$22354321$, $12232110$, $12233211$, $12343321$	$22354321$, $12232110$, $12233211$, $12343321$
10	$22354321$, $12232100$, $12233221$, $12343321$	$22354321$, $12233221$, $12343321$	11	$22354321$, $12232110$, $12232211$, $12344321$	$22354321$, $12232110$, $12232211$, $12344321$	12	$22354321$, $12232100$, $12232221$, $12344321$	$22354321$, $12344321$
13	$22454321$, $12232110$, $12233211$, $12243321$	$12232110$, $12233211$, $12243321$	14	$22454321$, $12232100$, $12233221$, $12243321$	$12233221$, $12243321$	15	$22454321$, $12232110$, $12232211$, $12244321$	$12232110$, $12232211$
16	$01010000$, $23454321$, $12244321$, $11232100$, $11232221$	$01010000$, $23454321$, $11232100$, $11232221$	17	$23465321$, $11121110$, $11122211$, $12343321$	$23465321$, $11121110$, $11122211$, $12343321$	18	$23465321$, $11121100$, $11122221$, $12343321$	$23465321$, $11121100$, $11122221$, $12343321$
19	$23464321$, $11122110$, $11122211$, $12344321$	$11122110$, $11122211$, $12344321$	20	$23464321$, $11122100$, $11122221$, $12344321$	$11122100$, $11122221$, $12344321$	21	$23465321$, $12243321$, $11221110$, $11222211$	$23465321$, $12243321$, $11221110$, $11222211$
22	$23465321$, $12243321$, $11221100$, $11222221$	$23465321$, $12243321$, $11221100$	23	$23464321$, $12244321$, $11222110$, $11222211$	$11222110$, $11222211$	24	$23464321$, $00000100$, $12244321$, $11222100$, $11222221$	$00000100$
25	$01000000$, $23465321$, $11121110$, $11122211$, $12343321$	$23465321$, $11121110$, $11122211$, $12343321$	26	$01000000$, $23465321$, $11121100$, $11122221$, $12343321$	$23465321$, $11121100$, $11122221$, $12343321$	27	$01000000$, $23464321$, $11122110$, $11122211$, $12344321$	$11122110$, $11122211$, $12344321$
28	$01000000$, $23464321$, $11122100$, $11122221$, $12344321$	$11122100$, $11122221$, $12344321$	29	$01000000$, $23465321$, $12243321$, $11221110$, $11222211$	$23465321$, $12243321$, $11221110$, $11222211$	30	$01000000$, $23465321$, $12243321$, $11221100$, $11222221$	$23465321$, $12243321$, $11221100$
31	$01000000$, $23464321$, $12244321$, $11222110$, $11222211$	$11222110$, $11222211$	32	$01000000$, $23464321$, $00000100$, $12244321$, $11222100$, $11222221$	$00000100$	33	$23465431$, $12232211$, $11233211$	$23465431$, $12232211$, $11233211$
34	$23465421$, $12232221$, $11233221$	$12232221$, $11233221$	35	$01011000$, $23465431$, $11122211$, $12343211$, $10111000$	$01011000$, $23465431$, $11122211$, $12343211$, $10111000$	36	$01011000$, $23465421$, $11122221$, $12343221$, $10111000$	$01011000$, $11122221$, $12343221$, $10111000$
37	$23465431$, $12232211$, $11221000$, $11233211$	$23465431$, $12232211$, $11233211$	38	$23465421$, $12232221$, $11221000$, $11233221$	$12232221$, $11233221$	39	$23465431$, $00011000$, $12233211$, $11221000$, $11232211$	$23465431$, $00011000$, $12233211$, $11232211$
40	$23465421$, $00011000$, $12233221$, $11221000$, $11232221$	$00011000$, $12233221$, $11232221$	41	$01010000$, $23465431$, $11111000$, $11122211$, $12343211$	$01010000$, $23465431$, $11111000$, $11122211$, $12343211$	42	$01010000$, $23465421$, $11111000$, $11122221$, $12343221$	$01010000$, $11111000$, $11122221$, $12343221$
43	$01111000$, $23465431$, $12232211$, $11121000$, $11233211$	$01111000$, $23465431$, $12232211$, $11121000$, $11233211$	44	$01111000$, $23465421$, $12232221$, $11121000$, $11233221$	$01111000$, $11121000$	45	$01010000$, $23465431$, $12233211$, $11221000$, $11232211$	$01010000$, $23465431$, $12233211$, $11232211$
46	$01010000$, $23465421$, $12233221$, $11221000$, $11232221$	$01010000$, $12233221$, $11232221$	47	$01011000$, $23465431$, $12232211$, $11221000$, $11233211$	$23465431$, $12232211$, $11233211$	48	$01010000$, $23465421$, $00011000$, $12233221$, $11221000$, $11232221$	$01010000$, $00011000$, $12233221$, $11232221$
49	$23465431$, $11121000$, $11122211$, $12343211$	$23465431$, $11121000$, $11122211$, $12343211$	50	$23465421$, $11121000$, $11122221$, $12343221$	$11121000$, $11122221$, $12343221$	51	$23465431$, $00001000$, $11121000$, $11122211$, $12343211$	$23465431$, $11121000$, $11122211$, $12343211$
52	$23465421$, $00001000$, $11121000$, $11122221$, $12343221$	$11121000$, $11122221$, $12343221$	53	$23465431$, $12243211$, $11221000$, $11222211$	$23465431$, $12243211$, $11222211$	54	$23465421$, $00000001$, $12243221$, $11221000$, $11222221$	$00000001$
55	$23465431$, $00001000$, $12243211$, $11221000$, $11222211$	$23465431$, $12243211$, $11222211$	56	$23465421$, $00001000$, $00000001$, $12243221$, $11221000$, $11222221$	$00000001$	57	$01000000$, $23465431$, $11121000$, $11122211$, $12343211$	$23465431$, $11121000$, $11122211$, $12343211$
58	$01000000$, $23465421$, $11121000$, $11122221$, $12343221$	$11121000$, $11122221$, $12343221$	59	$01000000$, $23465431$, $00001000$, $11121000$, $11122211$, $12343211$	$23465431$, $11121000$, $11122211$, $12343211$	60	$01000000$, $23465421$, $00001000$, $11121000$, $11122221$, $12343221$	$11121000$, $11122221$, $12343221$
61	$01000000$, $23465431$, $12243211$, $11221000$, $11222211$	$23465431$, $12243211$, $11222211$	62	$01000000$, $23465421$, $00000001$, $12243221$, $11221000$, $11222221$	$00000001$	63	$01000000$, $23465431$, $00001000$, $12243211$, $11221000$, $11222211$	$23465431$, $12243211$, $11222211$
64	$01000000$, $23465421$, $00001000$, $00000001$, $12243221$, $11221000$, $11222221$	$00000001$	65	$23354321$, $22454321$	$23354321$, $22454321$	66	$23354321$, $01122221$, $22454321$	$23354321$, $22454321$
67	$23354321$, $01122110$, $01122211$, $22454321$	$23354321$, $01122110$, $01122211$, $22454321$	68	$23354321$, $01122100$, $01122221$, $22454321$	$23354321$, $22454321$	69	$01110000$, $01122100$, $23465431$, $00111111$, $22343211$	$01110000$, $01122100$, $23465431$, $00111111$, $22343211$
70	$01110000$, $23465321$, $01122221$, $00111100$, $22343321$	$01110000$, $23465321$, $00111100$, $22343321$	71	$23454321$, $01122110$, $01122211$, $00110000$, $22354321$	$23454321$, $01122110$, $01122211$, $00110000$, $22354321$	72	$23454321$, $01122100$, $01122221$, $00110000$, $22354321$	$23454321$, $00110000$, $22354321$
73	$01010000$, $01111100$, $01122111$, $23465431$, $22343211$	$01010000$, $01111100$, $01122111$, $23465431$, $22343211$	74	$01010000$, $01111100$, $23465321$, $01122221$, $22343321$	$01010000$, $01111100$, $23465321$, $22343321$	75	$01010000$, $23454321$, $01122110$, $01122211$, $22354321$	$01010000$, $23454321$, $01122110$, $01122211$, $22354321$
76	$01010000$, $23454321$, $01122100$, $01122221$, $22354321$	$01010000$, $23454321$, $22354321$	77	$23354321$, $01111000$, $01121110$, $01122211$, $22454321$	$01111000$, $01121110$, $01122211$	78	$23354321$, $01111000$, $01121100$, $01122221$, $22454321$	$01111000$, $01121100$
79	$23354321$, $01110000$, $01122110$, $01122211$, $22454321$	$01122110$, $01122211$	80	$01010000$, $23454321$, $01122100$, $01122221$, $00110000$, $22354321$	$01010000$, $23454321$, $00110000$, $22354321$	81	$01121100$, $01122111$, $23465431$, $22343211$	$01121100$, $01122111$, $23465431$, $22343211$
82	$01121100$, $23465321$, $01122221$, $22343321$	$01121100$, $23465321$, $22343321$	83	$23464321$, $01122110$, $01122211$, $22344321$	$01122110$, $01122211$	84	$01121100$, $23465321$, $01122221$, $00001100$, $22343321$	$01121100$, $23465321$, $00001100$, $22343321$
85	$01121100$, $01122111$, $23465431$, $00100000$, $22343211$	$01121100$, $01122111$, $23465431$, $22343211$	86	$01121100$, $23465321$, $01122221$, $00100000$, $22343321$	$01121100$, $23465321$, $22343321$	87	$23464321$, $01122110$, $01122211$, $00100000$, $22344321$	$01122110$, $01122211$
88	$01121100$, $23465321$, $01122221$, $00100000$, $00001100$, $22343321$	$01121100$, $23465321$, $00001100$, $22343321$	89	$01000000$, $01121100$, $01122111$, $23465431$, $22343211$	$01121100$, $01122111$, $23465431$, $22343211$	90	$01000000$, $01121100$, $23465321$, $01122221$, $22343321$	$01121100$, $23465321$, $22343321$
91	$01000000$, $23464321$, $01122110$, $01122211$, $22344321$	$01122110$, $01122211$	92	$01000000$, $01121100$, $23465321$, $01122221$, $00001100$, $22343321$	$01121100$, $23465321$, $00001100$, $22343321$	93	$01000000$, $01121100$, $01122111$, $23465431$, $00100000$, $22343211$	$01121100$, $01122111$, $23465431$, $22343211$
94	$01000000$, $01121100$, $23465321$, $01122221$, $00100000$, $22343321$	$01121100$, $23465321$, $22343321$	95	$01000000$, $23464321$, $01122110$, $01122211$, $00100000$, $22344321$	$01122110$, $01122211$	96	$01000000$, $01121100$, $23465321$, $01122221$, $00100000$, $00001100$, $22343321$	$01121100$, $23465321$, $00001100$, $22343321$
97	$01122211$, $23465431$, $22343211$	$01122211$, $23465431$, $22343211$	98	$01000000$, $23465421$, $01122221$, $22343221$	$01000000$	99	$01011000$, $01122211$, $23465431$, $00111000$, $22343211$	$01011000$, $01122211$, $23465431$, $00111000$, $22343211$
100	$01011000$, $23465421$, $01122221$, $00111000$, $22343221$	$01011000$, $00111000$	101	$01110000$, $01122211$, $23465431$, $00111000$, $22343211$	$01110000$, $01122211$, $23465431$, $00111000$, $22343211$	102	$01110000$, $23465421$, $01122221$, $00111000$, $22343221$	$01110000$, $00111000$
103	$01111000$, $01122211$, $23465431$, $00110000$, $00011000$, $22343211$	$01111000$, $01122211$, $23465431$, $00110000$, $00011000$, $22343211$	104	$01111000$, $23465421$, $01122221$, $00110000$, $00011000$, $22343221$	$01111000$, $00110000$, $00011000$	105	$01010000$, $01111000$, $01122211$, $23465431$, $22343211$	$01010000$, $01111000$, $01122211$, $23465431$, $22343211$
106	$01010000$, $01111000$, $23465421$, $01122221$, $22343221$	$01010000$, $01111000$	107	$01010000$, $01111000$, $01122211$, $23465431$, $00011000$, $22343211$	$01010000$, $01111000$, $01122211$, $23465431$, $00011000$, $22343211$	108	$01010000$, $01111000$, $23465421$, $01122221$, $00011000$, $22343221$	$01010000$, $01111000$, $00011000$
109	$01010000$, $01111000$, $01122211$, $23465431$, $00110000$, $22343211$	$01010000$, $01111000$, $01122211$, $23465431$, $00110000$, $22343211$	110	$01010000$, $01111000$, $23465421$, $01122221$, $00110000$, $22343221$	$01010000$, $01111000$, $00110000$	111	$01011000$, $01110000$, $01122211$, $23465431$, $00111000$, $22343211$	$01122211$, $23465431$, $22343211$
112	$01010000$, $01111000$, $23465421$, $01122221$, $00110000$, $00011000$, $22343221$	$01010000$, $01111000$, $00110000$, $00011000$	113	$01121000$, $01122211$, $23465431$, $22343211$	$01122211$, $23465431$, $22343211$	114	$01121000$, $01122211$, $23465431$, $00000011$, $22343211$	$01122211$, $23465431$, $00000011$, $22343211$
115	$01121000$, $01122211$, $23465431$, $00001000$, $22343211$	$01122211$, $23465431$, $22343211$	116	$01121000$, $01122211$, $23465431$, $00001000$, $00000011$, $22343211$	$01122211$, $23465431$, $00000011$, $22343211$	117	$01121000$, $01122211$, $23465431$, $00100000$, $22343211$	$01122211$, $23465431$, $22343211$
118	$01121000$, $01122211$, $23465431$, $00100000$, $00000011$, $22343211$	$01122211$, $23465431$, $00000011$, $22343211$	119	$01121000$, $01122211$, $23465431$, $00100000$, $00001000$, $22343211$	$01122211$, $23465431$, $22343211$	120	$01121000$, $01122211$, $23465431$, $00100000$, $00001000$, $00000011$, $22343211$	$01122211$, $23465431$, $00000011$, $22343211$
121	$01000000$, $01121000$, $01122211$, $23465431$, $22343211$	$01122211$, $23465431$, $22343211$	122	$01000000$, $01121000$, $01122211$, $23465431$, $00000011$, $22343211$	$01122211$, $23465431$, $00000011$, $22343211$	123	$01000000$, $01121000$, $01122211$, $23465431$, $00001000$, $22343211$	$01122211$, $23465431$, $22343211$
124	$01000000$, $01121000$, $01122211$, $23465431$, $00001000$, $00000011$, $22343211$	$01122211$, $23465431$, $00000011$, $22343211$	125	$01000000$, $01121000$, $01122211$, $23465431$, $00100000$, $22343211$	$01122211$, $23465431$, $22343211$	126	$01000000$, $01121000$, $01122211$, $23465431$, $00100000$, $00000011$, $22343211$	$01122211$, $23465431$, $00000011$, $22343211$
127	$01000000$, $01121000$, $01122211$, $23465431$, $00100000$, $00001000$, $22343211$	$01122211$, $23465431$, $22343211$	128	$01000000$, $01121000$, $01122211$, $23465431$, $00100000$, $00001000$, $00000011$, $22343211$	$01122211$, $23465431$, $00000011$, $22343211$	129	$01000000$, $23465432$	$01000000$
130	$01011110$, $23465432$, $12343210$, $10111110$	$01011110$, $12343210$, $10111110$	131	$01011100$, $23465432$, $11122110$, $12343210$, $10111100$	$01011100$, $11122110$, $12343210$, $10111100$	132	$01011110$, $23465432$, $11122100$, $12343210$, $10111110$	$01011110$, $11122100$, $12343210$, $10111110$
133	$23465432$, $12232100$, $11221110$, $11233210$	$12232100$, $11221110$, $11233210$	134	$23465432$, $00011110$, $12232110$, $11222210$, $11232100$	$00011110$, $12232110$, $11222210$, $11232100$	135	$23465432$, $00011100$, $12233210$, $11221100$, $11232110$	$00011100$, $12233210$, $11221100$, $11232110$
136	$23465432$, $00011110$, $12233210$, $11221110$, $11232100$	$00011110$, $12233210$, $11232100$	137	$01010000$, $23465432$, $11111100$, $11122110$, $12343210$	$01010000$, $11111100$, $11122110$, $12343210$	138	$01111110$, $23465432$, $12232100$, $11122110$, $11232210$	$01111110$, $11122110$, $11232210$
139	$01111100$, $23465432$, $12232110$, $11121100$, $11233210$	$01111100$, $12232110$, $11121100$	140	$01111110$, $23465432$, $12232100$, $11121110$, $11233210$	$01111110$, $11121110$	141	$01010000$, $23465432$, $12233210$, $11221100$, $11232110$	$01010000$, $12233210$, $11221100$, $11232110$
142	$01011110$, $23465432$, $12232100$, $11222110$, $11232210$	$11222110$, $11232210$	143	$01011100$, $23465432$, $12232110$, $11221100$, $11233210$	$01011100$, $12232110$, $11221100$	144	$01010000$, $23465432$, $00011110$, $12233210$, $11221110$, $11232100$	$01010000$, $00011110$, $12233210$, $11232100$
145	$23465432$, $11121100$, $11122110$, $12343210$	$11121100$, $11122110$, $12343210$	146	$23465432$, $00000110$, $11121100$, $11122110$, $12343210$	$00000110$, $11121100$, $11122110$, $12343210$	147	$23465432$, $00001100$, $11121100$, $11122110$, $12343210$	$00001100$, $11121100$, $11122110$, $12343210$
148	$23465432$, $00001110$, $11121110$, $11122100$, $12343210$	$11121110$, $11122100$, $12343210$	149	$23465432$, $12243210$, $11221100$, $11222110$	$11221100$, $11222110$	150	$23465432$, $00000110$, $12243210$, $11221100$, $11222110$	$00000110$, $11221100$, $11222110$
151	$23465432$, $00001100$, $12243210$, $11221100$, $11222110$	$00001100$, $11221100$, $11222110$	152	$23465432$, $00001100$, $00000110$, $12243210$, $11221100$, $11222110$	$00001100$, $00000110$, $11221100$, $11222110$	153	$01000000$, $23465432$, $11121100$, $11122110$, $12343210$	$11121100$, $11122110$, $12343210$
154	$01000000$, $23465432$, $00000110$, $11121100$, $11122110$, $12343210$	$00000110$, $11121100$, $11122110$, $12343210$	155	$01000000$, $23465432$, $00001100$, $11121100$, $11122110$, $12343210$	$00001100$, $11121100$, $11122110$, $12343210$	156	$01000000$, $23465432$, $00001110$, $11121110$, $11122100$, $12343210$	$11121110$, $11122100$, $12343210$
157	$01000000$, $23465432$, $12243210$, $11221100$, $11222110$	$11221100$, $11222110$	158	$01000000$, $23465432$, $00000110$, $12243210$, $11221100$, $11222110$	$00000110$, $11221100$, $11222110$	159	$01000000$, $23465432$, $00001100$, $12243210$, $11221100$, $11222110$	$00001100$, $11221100$, $11222110$
160	$01000000$, $23465432$, $00001100$, $00000110$, $12243210$, $11221100$, $11222110$	$00001100$, $00000110$, $11221100$, $11222110$	161	$23465432$, $12232210$, $11233210$	$12232210$, $11233210$	162	$23465432$, $00000010$, $12232210$, $11233210$	$12232210$, $11233210$
163	$01011000$, $23465432$, $11122210$, $12343210$, $10111000$	$01011000$, $11122210$, $12343210$, $10111000$	164	$01011000$, $23465432$, $00000010$, $11122210$, $12343210$, $10111000$	$01011000$, $11122210$, $12343210$, $10111000$	165	$23465432$, $12232210$, $11221000$, $11233210$	$12232210$, $11233210$
166	$23465432$, $00000010$, $12232210$, $11221000$, $11233210$	$12232210$, $11233210$	167	$23465432$, $00011000$, $12233210$, $11221000$, $11232210$	$00011000$, $12233210$, $11232210$	168	$23465432$, $00011000$, $00000010$, $12233210$, $11221000$, $11232210$	$00011000$, $12233210$, $11232210$
169	$01010000$, $23465432$, $11111000$, $11122210$, $12343210$	$01010000$, $11111000$, $11122210$, $12343210$	170	$01010000$, $23465432$, $00000010$, $11111000$, $11122210$, $12343210$	$01010000$, $11111000$, $11122210$, $12343210$	171	$01111000$, $23465432$, $12232210$, $11121000$, $11233210$	$01111000$, $11121000$
172	$01111000$, $23465432$, $00000010$, $12232210$, $11121000$, $11233210$	$01111000$, $11121000$	173	$01010000$, $23465432$, $12233210$, $11221000$, $11232210$	$01010000$, $12233210$, $11232210$	174	$01010000$, $23465432$, $00000010$, $12233210$, $11221000$, $11232210$	$01010000$, $12233210$, $11232210$
175	$01010000$, $23465432$, $00011000$, $12233210$, $11221000$, $11232210$	$01010000$, $00011000$, $12233210$, $11232210$	176	$01010000$, $23465432$, $00011000$, $00000010$, $12233210$, $11221000$, $11232210$	$01010000$, $00011000$, $12233210$, $11232210$	177	$23465432$, $11121000$, $11122210$, $12343210$	$11121000$, $11122210$, $12343210$
178	$23465432$, $00000010$, $11121000$, $11122210$, $12343210$	$11121000$, $11122210$, $12343210$	179	$23465432$, $00001000$, $11121000$, $11122210$, $12343210$	$11121000$, $11122210$, $12343210$	180	$23465432$, $00001000$, $00000010$, $11121000$, $11122210$, $12343210$	$11121000$, $11122210$, $12343210$
181	$23465432$, $12243210$, $10000000$, $11221000$, $11222210$	$10000000$	182	$23465432$, $00000010$, $12243210$, $10000000$, $11221000$, $11222210$	$10000000$	183	$23465432$, $00001000$, $12243210$, $10000000$, $11221000$, $11222210$	$10000000$
184	$23465432$, $00001000$, $00000010$, $12243210$, $10000000$, $11221000$, $11222210$	$10000000$	185	$01000000$, $23465432$, $11121000$, $11122210$, $12343210$	$11121000$, $11122210$, $12343210$	186	$01000000$, $23465432$, $00000010$, $11121000$, $11122210$, $12343210$	$11121000$, $11122210$, $12343210$
187	$01000000$, $23465432$, $00001000$, $11121000$, $11122210$, $12343210$	$11121000$, $11122210$, $12343210$	188	$01000000$, $23465432$, $00001000$, $00000010$, $11121000$, $11122210$, $12343210$	$11121000$, $11122210$, $12343210$	189	$01000000$, $23465432$, $12243210$, $10000000$, $11221000$, $11222210$	$10000000$
190	$01000000$, $23465432$, $00000010$, $12243210$, $10000000$, $11221000$, $11222210$	$10000000$	191	$01000000$, $23465432$, $00001000$, $12243210$, $10000000$, $11221000$, $11222210$	$10000000$	192	$01000000$, $23465432$, $00001000$, $00000010$, $12243210$, $10000000$, $11221000$, $11222210$	$10000000$
193	$01000000$, $23465432$, $22343210$	$01000000$	194	$01011110$, $23465432$, $00111110$, $22343210$	$01011110$, $00111110$	195	$01011100$, $01122110$, $23465432$, $00111100$, $22343210$	$01011100$, $01122110$, $00111100$
196	$01011110$, $01122100$, $23465432$, $00111110$, $22343210$	$01011110$, $00111110$	197	$01110000$, $01122100$, $23465432$, $00111110$, $22343210$	$01110000$, $01122100$, $00111110$	198	$01110000$, $01122110$, $23465432$, $00111100$, $00000110$, $22343210$	$01110000$, $01122110$, $00111100$, $00000110$
199	$01111100$, $01122110$, $23465432$, $00110000$, $00011100$, $22343210$	$01111100$, $01122110$, $00110000$, $00011100$	200	$01111110$, $01122100$, $23465432$, $00110000$, $00011110$, $22343210$	$01111110$, $00110000$, $00011110$	201	$01010000$, $01111100$, $01122110$, $23465432$, $22343210$	$01010000$, $01111100$, $01122110$
202	$01010000$, $01111100$, $01122110$, $23465432$, $00000110$, $22343210$	$01010000$, $01111100$, $01122110$, $00000110$	203	$01010000$, $01111100$, $01122110$, $23465432$, $00011100$, $22343210$	$01010000$, $01111100$, $01122110$, $00011100$	204	$01010000$, $01111110$, $01122100$, $23465432$, $00011110$, $22343210$	$01010000$, $01111110$, $00011110$
205	$01010000$, $01111100$, $01122110$, $23465432$, $00110000$, $22343210$	$01010000$, $01111100$, $01122110$, $00110000$	206	$01011110$, $01111000$, $01121100$, $23465432$, $00111110$, $22343210$	$01111000$, $01121100$	207	$01011100$, $01110000$, $01122110$, $23465432$, $00111100$, $22343210$	$01011100$, $01122110$, $00111100$
208	$01010000$, $01111110$, $01122100$, $23465432$, $00110000$, $00011110$, $22343210$	$01010000$, $01111110$, $00110000$, $00011110$	209	$01121100$, $01122110$, $23465432$, $22343210$	$01121100$, $01122110$	210	$01121100$, $01122110$, $23465432$, $00000110$, $22343210$	$01121100$, $01122110$, $00000110$
211	$01121100$, $01122110$, $23465432$, $00001100$, $22343210$	$01121100$, $01122110$, $00001100$	212	$01121100$, $01122110$, $23465432$, $00001100$, $00000110$, $22343210$	$01121100$, $01122110$, $00001100$, $00000110$	213	$01121100$, $01122110$, $23465432$, $00100000$, $22343210$	$01121100$, $01122110$
214	$01121100$, $01122110$, $23465432$, $00100000$, $00000110$, $22343210$	$01121100$, $01122110$, $00000110$	215	$01121100$, $01122110$, $23465432$, $00100000$, $00001100$, $22343210$	$01121100$, $01122110$, $00001100$	216	$01121100$, $01122110$, $23465432$, $00100000$, $00001100$, $00000110$, $22343210$	$01121100$, $01122110$, $00001100$, $00000110$
217	$01000000$, $01121100$, $01122110$, $23465432$, $22343210$	$01121100$, $01122110$	218	$01000000$, $01121100$, $01122110$, $23465432$, $00000110$, $22343210$	$01121100$, $01122110$, $00000110$	219	$01000000$, $01121100$, $01122110$, $23465432$, $00001100$, $22343210$	$01121100$, $01122110$, $00001100$
220	$01000000$, $01121100$, $01122110$, $23465432$, $00001100$, $00000110$, $22343210$	$01121100$, $01122110$, $00001100$, $00000110$	221	$01000000$, $01121100$, $01122110$, $23465432$, $00100000$, $22343210$	$01121100$, $01122110$	222	$01000000$, $01121100$, $01122110$, $23465432$, $00100000$, $00000110$, $22343210$	$01121100$, $01122110$, $00000110$
223	$01000000$, $01121100$, $01122110$, $23465432$, $00100000$, $00001100$, $22343210$	$01121100$, $01122110$, $00001100$	224	$01000000$, $01121100$, $01122110$, $23465432$, $00100000$, $00001100$, $00000110$, $22343210$	$01121100$, $01122110$, $00001100$, $00000110$	225	$01000000$, $01122210$, $23465432$, $22343210$	$01000000$
226	$01000000$, $01122210$, $23465432$, $00000010$, $22343210$	$01000000$	227	$01011000$, $01122210$, $23465432$, $00111000$, $22343210$	$01011000$, $00111000$	228	$01011000$, $01122210$, $23465432$, $00111000$, $00000010$, $22343210$	$01011000$, $00111000$
229	$01110000$, $01122210$, $23465432$, $00111000$, $22343210$	$01110000$, $00111000$	230	$01110000$, $01122210$, $23465432$, $00111000$, $00000010$, $22343210$	$01110000$, $00111000$	231	$01111000$, $01122210$, $23465432$, $00110000$, $00011000$, $22343210$	$01111000$, $00110000$, $00011000$
232	$01111000$, $01122210$, $23465432$, $00110000$, $00011000$, $00000010$, $22343210$	$01111000$, $00110000$, $00011000$	233	$01010000$, $01111000$, $01122210$, $23465432$, $22343210$	$01010000$, $01111000$	234	$01010000$, $01111000$, $01122210$, $23465432$, $00000010$, $22343210$	$01010000$, $01111000$
235	$01010000$, $01111000$, $01122210$, $23465432$, $00011000$, $22343210$	$01010000$, $01111000$, $00011000$	236	$01010000$, $01111000$, $01122210$, $23465432$, $00011000$, $00000010$, $22343210$	$01010000$, $01111000$, $00011000$	237	$01010000$, $01111000$, $01122210$, $23465432$, $00110000$, $22343210$	$01010000$, $01111000$, $00110000$
238	$01010000$, $01111000$, $01122210$, $23465432$, $00110000$, $00000010$, $22343210$	$01010000$, $01111000$, $00110000$	239	$01010000$, $01111000$, $01122210$, $23465432$, $00110000$, $00011000$, $22343210$	$01010000$, $01111000$, $00110000$, $00011000$	240	$01010000$, $01111000$, $01122210$, $23465432$, $00110000$, $00011000$, $00000010$, $22343210$	$01010000$, $01111000$, $00110000$, $00011000$

Equations

$$\begin{array}{rl}& \text{i) J is centrally generated};\\ & \text{ii) all scalars cβ, β∈B∖Δ, are nonzero};\\ & \text{iii) J=J(f) for a Kostant form f}.\end{array}$$

$$\begin{array}{rl}{A}_{n-1}^{+}& =\{{ϵ}_i-{ϵ}_j,\text{leavevmode}1\le i<j\le n\},\\ B_n^{+}& =\{{ϵ}_i-{ϵ}_j,\text{leavevmode}1\le i<j\le n\}\cup \{{ϵ}_i+{ϵ}_j,\text{leavevmode}1\le i<j\le n\}\cup \{{ϵ}_i,\text{leavevmode}1\le i\le n\},\\ C_n^{+}& =\{{ϵ}_i-{ϵ}_j,\text{leavevmode}1\le i<j\le n\}\cup \{{ϵ}_i+{ϵ}_j,\text{leavevmode}1\le i<j\le n\}\cup \{2{ϵ}_i,\text{leavevmode}1\le i\le n\},\\ D_n^{+}& =\{{ϵ}_i-{ϵ}_j,\text{leavevmode}1\le i<j\le n\}\cup \{{ϵ}_i+{ϵ}_j,\text{leavevmode}1\le i<j\le n\},\\ E_6^{+}& =\{\pm {ϵ}_i+{ϵ}_j,\text{leavevmode}1\le i<j\le 5\}\cup \left\{\frac{1}{2}\left({ϵ}_8-{ϵ}_7-{ϵ}_6+\sum _{i=1}^5(-1{)}^{\nu (i)}{ϵ}_i\right),\text{leavevmode}\sum _{i=1}^5\nu (i)\text{ is even}\right\},\\ E_7^{+}& =\{\pm {ϵ}_i+{ϵ}_j,\text{leavevmode}1\le i<j\le 6\}\cup \{{ϵ}_8-{ϵ}_7\}\cup \left\{\frac{1}{2}\left({ϵ}_8-{ϵ}_7+\sum _{i=1}^6(-1{)}^{\nu (i)}{ϵ}_i\right),\text{leavevmode}\sum _{i=1}^6\nu (i)\text{ is even}\right\},\\ E_8^{+}& =\{\pm {ϵ}_i+{ϵ}_j,1\le i<j\le 8\}\cup \left\{\frac{1}{2}\left({ϵ}_8+\sum _{i=1}^7(-1{)}^{\nu (i)}{ϵ}_i\right),\text{leavevmode}\sum _{i=1}^7\nu (i)\text{ is even}\right\},\\ F_4^{+}& =\{{ϵ}_i\pm {ϵ}_j,\text{leavevmode}1\le i<j\le 4\}\cup \{({ϵ}_1\pm {ϵ}_2\pm {ϵ}_3\pm {ϵ}_4)/2\}\cup \{{ϵ}_i,1\le i\le 4\},\\ G_2^{+}& =\{{ϵ}_1-{ϵ}_2,\text{leavevmode}-2{ϵ}_1+{ϵ}_2+{ϵ}_3,\text{leavevmode}-{ϵ}_1+{ϵ}_3,-{ϵ}_2+{ϵ}_3,\text{leavevmode}{ϵ}_1-2{ϵ}_2+{ϵ}_3,\text{leavevmode}-{ϵ}_1-{ϵ}_2+2{ϵ}_3\}.\end{array}$$

$$\Delta =\{\begin{array}{ll}{\bigcup }_{i=1}^{n-1}\{{\alpha }_i={ϵ}_i-{ϵ}_{i+1}\}& \text{for }{A}_{n-1},\\ {\bigcup }_{i=1}^{n-1}\{{\alpha }_i={ϵ}_i-{ϵ}_{i+1}\}\cup \{{\alpha }_n={ϵ}_n\}& \text{for }{B}_n,\\ {\bigcup }_{i=1}^{n-1}\{{\alpha }_i={ϵ}_i-{ϵ}_{i+1}\}\cup \{{\alpha }_n=2{ϵ}_n\}& \text{for }{C}_n,\\ {\bigcup }_{i=1}^{n-1}\{{\alpha }_i={ϵ}_i-{ϵ}_{i+1}\}\cup \{{\alpha }_n={ϵ}_{n-1}+{ϵ}_n\}& \text{for }{D}_n,\\ \{{\alpha }_1=({ϵ}_1+{ϵ}_8-{\sum }_{k=2}^7{ϵ}_k)/2,& \\ {\alpha }_2={ϵ}_1+{ϵ}_2\}\cup {\bigcup }_{i=1}^4\{{\alpha }_{i+2}={ϵ}_{i+1}-{ϵ}_i\}& \text{for }{E}_6,\\ \{{\alpha }_1=({ϵ}_1+{ϵ}_8-{\sum }_{k=2}^7{ϵ}_k)/2,& \\ {\alpha }_2={ϵ}_1+{ϵ}_2\}\cup {\bigcup }_{i=1}^5\{{\alpha }_{i+1}={ϵ}_{i+1}-{ϵ}_i\}& \text{for }{E}_7,\\ \{{\alpha }_1=({ϵ}_1+{ϵ}_8-{\sum }_{k=2}^7{ϵ}_k)/2,& \\ {\alpha }_2={ϵ}_1+{ϵ}_2\}\cup {\bigcup }_{i=1}^6\{{\alpha }_{i+2}={ϵ}_{i+1}-{ϵ}_i\}& \text{for }{E}_8,\\ \{{\alpha }_1={ϵ}_2-{ϵ}_3,\text{leavevmode}{\alpha }_2={ϵ}_3-{ϵ}_4,\text{leavevmode}{\alpha }_3={ϵ}_4,& \\ {\alpha }_4=({ϵ}_1-{ϵ}_2-{ϵ}_3-{ϵ}_4)/2\}& \text{for }{F}_4,\\ \{{\alpha }_1={ϵ}_1-{ϵ}_2,\text{leavevmode}{\alpha }_2=-2{ϵ}_1+{ϵ}_2+{ϵ}_3\}& \text{for }{G}_2.\end{array}$$

$$\mathcal{B}=\{\begin{array}{ll}{\bigcup }_{i=1}^{[n/2]}\{{\beta }_i={ϵ}_i-{ϵ}_{n-i+1}\}& \text{for }{A}_{n-1},\\ {\bigcup }_{i=1}^{n/2}\{{\beta }_{2i-1}={ϵ}_{2i-1}+{ϵ}_{2i},\text{leavevmode}{\beta }_{2i}={ϵ}_{2i-1}-{ϵ}_{2i}\}& \text{for }{B}_n,\text{leavevmode}n\text{ even},\\ {\bigcup }_{i=1}^{[n/2]}\{{\beta }_{2i-1}={ϵ}_{2i-1}+{ϵ}_{2i},\text{leavevmode}{\beta }_{2i}={ϵ}_{2i-1}-{ϵ}_{2i}\}\cup \{{\beta }_n={ϵ}_n\}& \text{for }{B}_n,\text{leavevmode}n\text{ odd},\\ {\bigcup }_{i=1}^n\{{\beta }_i=2{ϵ}_i\}& \text{for }{C}_n,\\ {\bigcup }_{i=1}^{[n/2]}\{{\beta }_{2i-1}={ϵ}_{2i-1}+{ϵ}_{2i},\text{leavevmode}{\beta }_{2i}={ϵ}_{2i-1}-{ϵ}_{2i}\}& \text{for }{D}_n,\\ \{{\beta }_1=({ϵ}_1+{ϵ}_2+{ϵ}_3+{ϵ}_4+{ϵ}_5-{ϵ}_6-{ϵ}_7+{ϵ}_8)/2,& \\ {\beta }_2=(-{ϵ}_1-{ϵ}_2-{ϵ}_3-{ϵ}_4+{ϵ}_5-{ϵ}_6-{ϵ}_7+{ϵ}_8)/2,& \\ {\beta }_3=-{ϵ}_1+{ϵ}_4,\text{leavevmode}{\beta }_4=-{ϵ}_2+{ϵ}_3\}& \text{for }{E}_6,\\ \{{\beta }_1=-{ϵ}_7+{ϵ}_8,\text{leavevmode}{\beta }_2={ϵ}_5+{ϵ}_6,\text{leavevmode}{\beta }_3={ϵ}_3+{ϵ}_4,{\beta }_4=-{ϵ}_5+{ϵ}_6,& \\ {\beta }_5={ϵ}_1+{ϵ}_2,{\beta }_6=-{ϵ}_1+{ϵ}_2,\text{leavevmode}{\beta }_7=-{ϵ}_3+{ϵ}_4\}& \text{for }{E}_7,\\ \{{\beta }_1={ϵ}_7+{ϵ}_8,\text{leavevmode}{\beta }_2=-{ϵ}_7+{ϵ}_8,\text{leavevmode}{\beta }_3={ϵ}_5+{ϵ}_6,\text{leavevmode}{\beta }_4={ϵ}_3+{ϵ}_4,& \\ \text{leavevmode}{\beta }_5=-{ϵ}_5+{ϵ}_6,\text{leavevmode}{\beta }_6={ϵ}_1+{ϵ}_2,\text{leavevmode}{\beta }_7=-{ϵ}_1+{ϵ}_2,\text{leavevmode}{\beta }_8=-{ϵ}_3+{ϵ}_4\}& \text{for }{E}_8,\\ \{{\beta }_1={ϵ}_1+{ϵ}_2,\text{leavevmode}{\beta }_2={ϵ}_1-{ϵ}_2,\text{leavevmode}{\beta }_3={ϵ}_3+{ϵ}_4,\text{leavevmode}{\beta }_4={ϵ}_3-{ϵ}_4\}& \text{for }{F}_4,\\ \{{\beta }_1=-{ϵ}_1-{ϵ}_2+2{ϵ}_3,\text{leavevmode}{\beta }_2={ϵ}_1-{ϵ}_2\}& \text{for }{G}_2.\end{array}$$

$$\sigma :S(\mathfrak{n})\to U(\mathfrak{n}),\text{leavevmode}{x}^k\mapsto x^k,\text{leavevmode}x\in \mathfrak{n},\text{leavevmode}k\in {\mathbb{Z}}_{\ge 0},$$

$${\xi }_{\beta },\text{leavevmode}\beta \in \mathcal{B},\text{ are algebraically independent generators of }Y(\mathfrak{n}),$$

$${\mu }_{{\beta }_i}\in ⟨{\beta }_1,\text{leavevmode}\dots ,\text{leavevmode}{\beta }_i{⟩}_{\mathbb{R}}\setminus ⟨{\beta }_1,\text{leavevmode}\dots ,\text{leavevmode}{\beta }_{i-1}{⟩}_{\mathbb{R}},$$

$$R=\left\{t={\sum }_{\beta \in \mathcal{B}}{t}_{\beta }{e}_{\beta }^{\ast },\text{leavevmode}{t}_{\beta }\in {\mathbb{C}}^{\times }\right\}$$

$${\xi }_{{\beta }_i}(t)={\prod }_{\beta \in \mathcal{B}}{t}_{\beta }^{r_{{\beta }_i}(\beta )},\text{leavevmode}1\le i\le |\mathcal{B}|,\text{ where }{r}_{{\beta }_i}(\beta )=({\mu }_{{\beta }_i},\beta )/(\beta ,\beta ).$$

$$\mathcal{D}:{\mathfrak{n}}^{\ast }\to \mathrm{Prim}U(\mathfrak{n}),\text{leavevmode}\lambda \mapsto J(\lambda ),$$

$$\begin{array}{rl}& \text{i) J is primitive;}\\ & \text{ii) J is maximal;}\\ & \text{iii) the center of U(n)/J is trivial;}\\ & \text{iv) U(n)/J is isomorphic to a Weyl algebra of finitely many variables.}\end{array}$$

$$\begin{array}{rl}& \text{i) J is centrally generated (or, equivalently, J=Jc)};\\ & \text{ii) the scalars cβ, β∈B∖Δ, are nonzero};\\ & \text{iii) J=J(fξ) for a Kostant form fξ∈n∗}.\end{array}$$

$$\begin{array}{rl}\tilde{\mathfrak{n}}& =⟨e_{\alpha },\text{leavevmode}\alpha \in {\Phi }^{+},\text{leavevmode}(\alpha ,{\beta }_1)=0{⟩}_{\mathbb{C}}=⟨e_{\alpha },\text{leavevmode}\alpha \in {\tilde{\Phi }}^{+}{⟩}_{\mathbb{C}},\\ \mathfrak{k}& =⟨e_{\alpha },\text{leavevmode}\alpha \in {\Phi }^{+},\text{leavevmode}(\alpha ,{\beta }_1)>0{⟩}_{\mathbb{C}},\end{array}$$

$$h\cdot r(y)=r([h,y])=1\otimes \overline{[h,y]}=h.(1\otimes \overline{y})=h.r(y),$$

$$h(1\otimes \theta (e_{\alpha }))=({\beta }_1+\alpha )(h)1\otimes \theta (\alpha )=\alpha (h)1\otimes \theta (e_{\alpha }),$$

$$h.r(e_{\alpha })=[h,e_{\alpha }]\otimes 1+1\otimes \alpha (h)\theta (e_{\alpha })=\alpha (h)r(e_{\alpha })=h\cdot r(e_{\alpha }).$$

$$U(\mathfrak{n})/J_c\cong (U(\mathfrak{n})/J_0)/r(J_c)\cong (U(\tilde{\mathfrak{n}})\otimes {\mathcal{A}}_s)/r(J_c)=(U(\tilde{\mathfrak{n}})/{\tilde{J}}_c)\otimes {\mathcal{A}}_s\cong {\mathcal{A}}_t\otimes {\mathcal{A}}_s\cong {\mathcal{A}}_{t+s}.$$

$$f_{D,\xi }=\sum _{\beta \in D}\xi (\beta )e_{\beta }^{\ast }\in {\mathfrak{n}}^{\ast }.$$

$${\tilde{f}}_{D,\xi }={\left(\mathrm{ad}\left({\sum }_{\beta \in D}\xi (\beta )e_{\beta }\right)\right)}^T\in {\mathfrak{u}}_{-}\cong {\mathfrak{u}}^{\ast }.$$

$${\mathrm{ad}}_{e_{{\beta }_0}}(h_{{\alpha }_0})=[e_{{\beta }_0,h_{{\alpha }_0}}]=-\frac{2({\alpha }_0,{\beta }_0)}{({\alpha }_0,{\alpha }_0)}{e}_{{\beta }_0},$$

$$f_{h_{{\alpha }_0},e_{\alpha }}=f_{e_{-\gamma },e_{{\beta }_0}}=0\text{ for all }{e}_{\alpha }{<}_t{e}_{{\beta }_0}\text{ and all }{e}_{-\gamma },\text{leavevmode}\alpha ,\gamma \in {\Phi }^{+}.$$

$$f_{h_{{\alpha }_0},e_{\beta }}=-\xi (\beta )\frac{2({\alpha }_0,\beta )}{({\alpha }_0,{\alpha }_0)}=0,$$

$$\begin{split}&\text{{i)} $D$ is linear independent {(}in fact{,} orthogonal{,} except cases $11${,} $12$ and $14$ for $F_{4}$ in Table {\ref{table:D_D'_F_4}}{)}};\\ &\text{{ii)} if $c_{\beta_{i}}=0$ for some $\beta_{i}\in\mathcal{B}$ then $\mu_{\beta_{i}}$ does not belong to the $\mathbb{Z}_{\geq 0}$-linear span of $D$};\\ &\text{{iii)} if $c_{\beta_{i}}\neq 0$ for some $\beta_{i}\in\mathcal{B}$ then $\mu_{\beta_{i}}$ belongs to the $\mathbb{Z}_{\geq_{0}}$-linear span of $D$};\\ &\text{{iv)} there exist $\alpha_{0}\in\Delta$ and $\beta_{0}\in D^{\prime}$ such that $(\alpha_{0},\beta_{0})\neq 0$ and $(\alpha_{0},\beta)=0$ for all $\beta\in D\setminus D^{\prime}$}.\\ \end{split}$$

$$\begin{array}{rl}D& =\{{\alpha }_3+{\alpha }_4,\text{leavevmode}{\alpha }_4+{\alpha }_5+{\alpha }_6,\text{leavevmode}{\alpha }_1+2{\alpha }_2+2{\alpha }_3+3{\alpha }_4+2{\alpha }_5+{\alpha }_6,\text{leavevmode}{\alpha }_1+{\alpha }_3+{\alpha }_4+{\alpha }_5\},\\ D^{\prime }& =\{{\alpha }_3+{\alpha }_4,\text{leavevmode}{\alpha }_4+{\alpha }_5+{\alpha }_6\}.\end{array}$$

$$\begin{array}{rl}& \text{i) J is centrally generated};\\ & \text{ii) the scalars cβ, β∈B∖Δ, are nonzero};\\ & \text{iii) J=J(fξ) for a Kostant form fξ∈n∗}.\end{array}$$

$$R=\left\{t={\sum }_{\beta \in \mathcal{B}}{t}_{\beta }{e}_{\beta }^{\ast },\text{leavevmode}{t}_{\beta }\in {\mathbb{C}}^{\times }\right\}$$

$${\xi }_{{\beta }_i}(t)=\prod _{\beta \in \mathcal{B}}{t}_{\beta }^{r_{{\beta }_i}(\beta )},\text{ where }{\mu }_{{\beta }_i}=\sum _{\beta \in \mathcal{B}}{r}_{{\beta }_i}(\beta )\beta ,$$

$${\mu }_{{\beta }_i}={\beta }_i+\sum _{j<i}{r}_{{\beta }_i}({\beta }_j){\beta }_j,\text{ so }{\xi }_{{\beta }_i}(t)=t_{{\beta }_i}\prod _{j<i}{t}_{{\beta }_j}^{r_{{\beta }_i}({\beta }_j)}.$$

$$f_{\xi }=\sum _{j=1}^m\xi ({\beta }_j)e_{{\beta }_j}^{\ast },\text{leavevmode}m=|\mathcal{B}|,$$

$${\xi }_{{\beta }_i}(f_{\xi })=\xi ({\beta }_i)\prod _{j<i}\xi ({\beta }_j{)}^{r_{{\beta }_i}({\beta }_j)}$$

Full text

Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1.

Mikhail V. Ignatyev

  

Aleksandr A. Shevchenko

[TABLE]

1. Introduction

The ground field is the field $\mathbb{C}$ of complex numbers. The theory of primitive ideals in enveloping algebras of Lie algebras can be considered as a part of the representation theory of Lie algebras. In general, it is impossible to classify all irreducible representations of Lie algebras (except in a few very special cases), while a classification of annihilators of irreducible representations, i.e., of primitive ideals, can be achieved in much greater generality. In the case when $\mathfrak{n}$ is a finite-dimensional nilpotent Lie algebra, the primitive ideals in the universal enveloping algebra $U(\mathfrak{n})$ can be described in terms of the Dixmier map assigning to any linear form $f\in\mathfrak{n}^{*}$ a primitive ideal $J(f)$ of $U(\mathfrak{n})$.

If $\mathfrak{n}$ is abelian, $J(f)$ is simply the annihilator of $f$. For a general finite-dimensional nilpotent Lie algebra $\mathfrak{n}$, the theory of primitive ideals retains many properties from the abelian case: in particular, $J(f)$ is always a maximal ideal and every primitive ideal in $U(\mathfrak{n})$ is of the form $J(f)$ for some $f\in\mathfrak{n}^{*}$. Moreover, $J(f)=J(f^{\prime})$ if and only if $f$ and $f^{\prime}$ belong to the same coadjoint orbit in $\mathfrak{n}^{*}$ [Di4, Theorem 6.2.4]. Note that, according to the Kostant–Kirillov orbit method, the set of all unitary irreducible representations of the Lie group $\exp\mathfrak{n}$ is also in one-to-one correspondence with the set of all coadjoint orbits in $\mathfrak{n}^{*}$ [Ki1]. Note also that there exists a natural topology, called the Jacobson topology, on the set $\mathrm{Prim}\,U(\mathfrak{n})$ of all primitive ideals of $U(\mathfrak{n})$. It turns out that $\mathrm{Prim}\,U(\mathfrak{n})$ is irreducible, and each primitive ideal from a certain (dense) open subset of $\mathrm{Prim}\,U(\mathfrak{n})$ is generated by its intersection with the center $Z(\mathfrak{n})$ of $U(\mathfrak{n})$; such an ideal is called centrally generated (see Section Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. for the details).

Suppose that $\mathfrak{n}$ is the nilradical of a Borel subalgebra $\mathfrak{b}$ of a complex finite-dimensional semisimple Lie algebra $\mathfrak{g}$. The description of $Z(\mathfrak{n})$ goes back to Dixmier, Joseph and Kostant. It turns out that $Z(\mathfrak{n})$ is a polynomial algebra whose generators are parametrized by the positive roots from the Kostant cascade $\mathcal{B}$, a certain strongly orthogonal subset of the set $\Phi^{+}$ of positive roots of $\mathfrak{g}$ with respect to $\mathfrak{b}$ [Ko2, Theorem 7] (see Section Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. for the precise definition). Let $\{e_{\alpha},\leavevmode\nobreak\ \alpha\in\Phi^{+}\}$ be a basis of $\mathfrak{n}$ consisting of root vectors. We say that a linear form $f\in\mathfrak{n}^{*}$ is a Kostant form if $f(e_{\alpha})=0$ for $\alpha\notin\mathcal{B}$ and $f(e_{\beta})\neq 0$ for $\beta\in\mathcal{B}\setminus\Delta$, where $\Delta$ is the set of simple roots. Note that the coadjoint orbit of a Kostant form has maximal possible dimension.

It was proved in [IP, Theorem 3.1] and [Ig1, Theorem 2.4] that, when $\Phi$ is of classical type (i.e., $\Phi=A_{n-1}$, $B_{n}$, $C_{n}$ or $D_{n}$), $J\in\mathrm{Prim}\,U(\mathfrak{n})$ is centrally generated if and only if $J=J(f)$ for a certain Kostant form $f$. In this paper, we prove that this fact is also true when $\Phi$ is of exceptional type, i.e., $\Phi=E_{6}$, $E_{7}$, $E_{8}$, $F_{4}$ or $G_{2}$. Namely, let $\Delta_{\beta}$, $\beta\in\mathcal{B}$, be the set of canonical generators of $Z(\mathfrak{n})$ (see Section Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1.). Let $J$ be a primitive ideal of $U(\mathfrak{n})$. Since $J$ is the annihilator of a simple $\mathfrak{n}$-module, given $\beta\in\mathcal{B}$, there exists unique $c_{\beta}\in\mathbb{C}$ such that $\Delta_{\beta}-c_{\beta}\in J$. Our main result, Theorem Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., claims that the following conditions are equivalent:

[TABLE]

If these conditions are satisfied, then $f$ can be explicitly reconstructed from $J$. As a corollary, we conclude that the same is true for arbitrary root system.

The paper is organized as follows. In Section Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., we briefly recall Kostant’s characterization of $Z(\mathfrak{n})$ and present a (more or less) explicit description of the canonical generators of $Z(\mathfrak{n})$ based on Panov’s work [Pa2]. Using this description, in Section Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. we prove that certain centrally generated ideals are primitive (in fact, it is the key ingredient in the proof of the main result, see Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1.). Section Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. is devoted to some particular classes of coadjoint orbits. Namely, we prove that certain orbits are disjoint, see Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1.. Finally, in Section 4, combining our results from two previous sections, we prove the main result, Theorem Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1.. As an immediate corollary, we obtain that the similar result is true for an arbitrary semisimple Lie algebra, see Theorem Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1..

Acknowledgments. We express our gratitude to A. Panov and I. Penkov for useful discussions. The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1.

2. The center of $U(\mathfrak{n})$

Let $G$ be a complex semisimple algebraic group, let $H$ be a Cartan subgroup of $G$, let $B$ be a Borel subgroup of $G$ containing $H$, and let $N$ be the unipotent radical of $B$. We denote by $\Phi$ the root system of $G$ with respect to $B$, and by $\Phi^{+}$ the set of positive roots with respect to $B$. Let $\mathfrak{g}$ (respectively, $\mathfrak{h}$, $\mathfrak{b}$ and $\mathfrak{n}$) be the Lie algebra of $G$ (respectively, of $H$, $B$ and $N$), so that $\mathfrak{b}=\mathfrak{h}\oplus\mathfrak{n}$ as vector spaces. The Lie algebra $\mathfrak{n}$ has a basis consisting of root vectors $e_{\alpha}$, $\alpha\in\Phi^{+}$. We denote the dual basis of the dual space $\mathfrak{n}^{*}$ by $\{e_{\alpha}^{*},\leavevmode\nobreak\ \alpha\in\Phi^{+}\}$.

Let $\mathbb{R}^{n}$ be the $n$-dimensional Euclidean space with the standard inner product $(\cdot,\cdot)$, and $\{\epsilon_{i}\}_{i=1}^{n}$ be the standard basis of $\mathbb{R}^{n}$. If $\Phi$ is irreducible then we identify $\Phi^{+}$ with the following subset of $\mathbb{R}^{n}$ [Bo]:

[TABLE]

Under this identification, the set $\Delta\subset\Phi^{+}$ of the simple roots has the following form:

[TABLE]

Recall that there exists a natural partial order on $\Phi$: by definition, $\alpha<\beta$ if $\beta-\alpha$ can be represented as a sum of positive roots. Denote by $\mathcal{B}$ the subset of $\Phi^{+}$ constructed by the following inductive procedure. Let $\mathcal{B}_{1}$ be the set consisting of the maximal roots of all irreducible components of $\Phi$. For $n\geq 2$, we denote $\Phi_{n}=\{\alpha\in\Phi\mid\alpha\perp\beta\text{ for all }\beta\in\mathcal{B}_{1}\cup\ldots\cup\mathcal{B}_{n-1}\}$, and set $\mathcal{B}_{n}$ to be the set of the maximal roots of all irreducible components of $\Phi_{n}$. Finally, we denote by $\mathcal{B}$ the union of all $\mathcal{B}_{n}$’s. Note that $\mathcal{B}$ is a maximal strongly orthogonal subset of $\Phi^{+}$, i.e., $\mathcal{B}$ is maximal with the property that if $\alpha,\leavevmode\nobreak\ \beta\in\mathcal{B}$ then neither $\alpha-\beta$ nor $\alpha+\beta$ belongs to $\Phi^{+}$.

**Definition 2.1. **We call $\mathcal{B}$ the Kostant cascade of orthogonal roots in $\Phi^{+}$.

Let $[n/2]$ be the largest integer not exceeding $n/2$. For irreducible $\Phi$, $\mathcal{B}$ has the following form:

[TABLE]

Denote by $U(\mathfrak{n})$ the enveloping algebra of $\mathfrak{n}$, and by $S(\mathfrak{n})$ the symmetric algebra of $\mathfrak{n}$. Then $\mathfrak{n}$ and $S(\mathfrak{n})$ are $B$-modules as $B$ normalizes $N$. Denote by $Z(\mathfrak{n})$ the center of $U(\mathfrak{n})$. It is well-known that the restriction of the symmetrization map

[TABLE]

to the algebra $Y(\mathfrak{n})$ of $N$-invariants in $S(\mathfrak{n})$ is an algebra isomorphism between $Y(\mathfrak{n})$ and $Z(\mathfrak{n})$.

We next present a canonical set of generators of $Z(\mathfrak{n})$ (or, equivalently, of $Y(\mathfrak{n})$), whose description goes back to Dixmier, Joseph and Kostant [Di3], [Jo1], [Ko1], [Ko2]. Let $\mathfrak{X}$ be the group of rational multiplicative characters of $H$, i.e., of algebraic group morphisms from $H$ to $\mathbb{C}^{\times}=\mathbb{C}\setminus\{0\}$. We can consider $\mathbb{Z}\Phi$, the $\mathbb{Z}$-linear span of $\Phi$, as a subgroup of $\mathfrak{X}$. Recall that a vector $\lambda\in\mathbb{R}^{n}$ is called a weight if $c(\alpha,\lambda):=2(\alpha,\lambda)/(\alpha,\alpha)$ is an integer for any $\alpha\in\Phi^{+}$. A weight $\lambda$ is called dominant if $c(\alpha,\lambda)\geq 0$ for all $\alpha\in\Phi^{+}$. An element $a$ of an $H$-module is called a weight vector if there exists $\nu\in\mathfrak{X}$ such that $h\cdot a=\nu(h)a$ for all $h\in H$. By [Ko2, Theorems 6, 7], every weight occurs in $Y(\mathfrak{n})$ with multiplicity at most 1. Furthermore, there exist unique (up to scalars) prime polynomials $\xi_{\beta}\in Y(\mathfrak{n})$, $\beta\in\mathcal{B}$, such that each $\xi_{\beta}$ is a weight polynomial of a dominant weight $\mu_{\beta}$ belonging to the $\mathbb{Z}$-linear span $\mathbb{Z}\mathcal{B}$ of $\mathcal{B}$. A remarkable fact is that

[TABLE]

so $Y(\mathfrak{n})$ and $Z(\mathfrak{n})$ are polynomial rings [Ko2, Theorem 7]. It turns out that the weights $\mu_{\beta}$’s have the following form [Pa2, Theorem 2.12].

Table 1. Weights $\mu_{\beta}$ for $\beta\in\mathcal{B}$

$\Phi=A_{n-1}$ $\mu_{\beta_{i}}=\epsilon_{1}+\ldots+\epsilon_{i}-\epsilon_{n-i+1}-\ldots-\epsilon_{n}$, $1\leq i\leq[n/2]$

$\Phi=B_{n}$ \mu_{\beta_{i}}=\begin{cases}2\epsilon_{1}+\ldots+2\epsilon_{i-1}&\text{for even i},\\ \epsilon_{1}+\ldots+\epsilon_{i}+\epsilon_{i+1}&\text{for odd i<n,}\\ \epsilon_{1}+\ldots+\epsilon_{i}&\text{for odd i=n,}\end{cases}

$\Phi=C_{n}$ $\mu_{\beta_{i}}=2\epsilon_{1}+\ldots+2\epsilon_{i}$, $1\leq i\leq n$

$\Phi=D_{n}$ \mu_{\beta_{i}}=\begin{cases}2\epsilon_{1}+\ldots+2\epsilon_{i-1}&\text{for even i<n},\\ \epsilon_{1}+\ldots+\epsilon_{n-1}-\epsilon_{n}&\text{for even i=n},\\ \epsilon_{1}+\ldots+\epsilon_{i+1}&\text{for odd i}\end{cases}

$\Phi=E_{6}$

$\mu_{\beta_{1}}=(\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4}+\epsilon_{5}-\epsilon_{6}-\epsilon_{7}+\epsilon_{8})/2,$

$\mu_{\beta_{2}}=\epsilon_{5}-\epsilon_{6}-\epsilon_{7}+\epsilon_{8},$

$\mu_{\beta_{3}}=(-\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+3\epsilon_{4}+3\epsilon_{5}-3\epsilon_{6}-3\epsilon_{7}+3\epsilon_{8})/2,$

$\mu_{\beta_{4}}=2\epsilon_{3}+2\epsilon_{4}+2\epsilon_{5}-2\epsilon_{6}-2\epsilon_{7}+2\epsilon_{8}$

$\Phi=E_{7}$

$\mu_{\beta_{1}}=-\epsilon_{7}+\epsilon_{8}$

$\mu_{\beta_{2}}=\epsilon_{5}+\epsilon_{6}-\epsilon_{7}+\epsilon_{8},$

$\mu_{\beta_{3}}=\epsilon_{3}+\epsilon_{4}+\epsilon_{5}+\epsilon_{6}-2\epsilon_{7}+2\epsilon_{8},$

$\mu_{\beta_{4}}=2\epsilon_{6}-\epsilon_{7}+\epsilon_{8},$

$\mu_{\beta_{4}}=\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{5}+\epsilon_{5}+\epsilon_{6}-2\epsilon_{7}+2\epsilon_{8},$

$\mu_{\beta_{5}}=-\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{6}+\epsilon_{5}+\epsilon_{6}-3\epsilon_{7}+3\epsilon_{8},$

$\mu_{\beta_{7}}=2\epsilon_{4}+2\epsilon_{5}+2\epsilon_{6}-3\epsilon_{7}+3\epsilon_{8}$

$\Phi=E_{8}$

$\mu_{\beta_{1}}=\epsilon_{7}+\epsilon_{8},\leavevmode\nobreak\ \mu_{\beta_{2}}=2\epsilon_{8},$

$\mu_{\beta_{3}}=\epsilon_{5}+\epsilon_{6}+\epsilon_{7}+3\epsilon_{8}$,

$\mu_{\beta_{4}}=\epsilon_{3}+\epsilon_{4}+\epsilon_{5}+\epsilon_{6}+\epsilon_{7}+5\epsilon_{8},$

$\mu_{\beta_{5}}=2\epsilon_{6}+2\epsilon_{7}+4\epsilon_{8},$

$\mu_{\beta_{6}}=\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4}+\epsilon_{5}+\epsilon_{6}+\epsilon_{7}+5\epsilon_{8},$

$\mu_{\beta_{7}}=-\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4}+\epsilon_{5}+\epsilon_{6}+\epsilon_{7}+7\epsilon_{8},$

$\mu_{\beta_{8}}=2\epsilon_{4}+2\epsilon_{5}+2\epsilon_{6}+2\epsilon_{7}+8\epsilon_{8}$

$\Phi=F_{4}$

$\mu_{\beta_{1}}=\epsilon_{1}+\epsilon_{2},\leavevmode\nobreak\ \mu_{\beta_{2}}=2\epsilon_{1},$

$\mu_{\beta_{3}}=3\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4},$

$\mu_{\beta_{4}}=4\epsilon_{1}+2\epsilon_{2}+2\epsilon_{3}$

$\Phi=G_{2}$ $\mu_{\beta_{1}}=-\epsilon_{1}-\epsilon_{2}+2\epsilon_{3},\leavevmode\nobreak\ \mu_{\beta_{2}}=-2\epsilon_{2}+2\epsilon_{3}$

For the sequel, we need to express the weights $\mu_{\beta}$’s as linear combination of simple roots. Such expressions are presented in Table Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1..

Table 2. Weights $\mu_{\beta}$, $\beta\in\mathcal{B}$, as linear combinations of simple roots

[TABLE]

Further, we also need to express the weights $\mu_{\beta_{i}}$’s as linear combinations of roots from $\mathcal{B}$ (it is possible since $\mu_{\beta_{i}}\in\mathbb{Z}\mathcal{B}$ for all $i$). Such expressions are presented in Table Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1..

Table 3. Weights $\mu_{\beta}$, $\beta\in\mathcal{B}$, as linear combinations of roots from $\mathcal{B}$

$\Phi=A_{n-1}$ $\mu_{\beta_{i}}=\beta_{1}+\ldots+\beta_{i}$, $1\leq i\leq[n/2]$

$\Phi=B_{n}$ \mu_{\beta_{i}}=\begin{cases}2\beta_{1}+2\beta_{3}+\ldots+2\beta_{i-2}+\beta_{i-1}+\beta_{i}&\text{for even i},\\ \beta_{1}+\beta_{3}\ldots+\beta_{i}&\text{for odd i<n}\\ \end{cases}

$\Phi=C_{n}$ $\mu_{\beta_{i}}=\beta_{1}+\ldots+\beta_{i}$, $1\leq i\leq n$

$\Phi=D_{n}$ \mu_{\beta_{i}}=\begin{cases}2\beta_{1}+2\beta_{3}+\ldots+2\beta_{i-2}+\beta_{i-1}+\beta_{i}&\text{for even i<n},\\ \beta_{1}+\beta_{3}+\ldots+\beta_{n-3}+\beta_{n}&\text{for even i=n},\\ \beta_{1}+\beta_{3}\ldots+\beta_{i}&\text{for odd i}\end{cases}

$\Phi=E_{6}$ $\mu_{\beta_{1}}=\beta_{1},\leavevmode\nobreak\ \mu_{\beta_{2}}=\beta_{1}+\beta_{2},\leavevmode\nobreak\ \mu_{\beta_{3}}=2\beta_{1}+\beta_{2}+\beta_{3},\leavevmode\nobreak\ \mu_{\beta_{4}}=3\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4}$

$\Phi=E_{7}$

$\mu_{\beta_{1}}=\beta_{1},\leavevmode\nobreak\ \mu_{\beta_{2}}=\beta_{1}+\beta_{2},$

$\mu_{\beta_{3}}=2\beta_{1}+\beta_{2}+\beta_{3},\leavevmode\nobreak\ \mu_{\beta_{4}}=\beta_{1}+\beta_{2}+\beta_{4}$

$\mu_{\beta_{5}}=2\beta_{1}+\beta_{2}+\beta_{3}+\beta_{5},\mu_{\beta_{6}}=3\beta_{1}+\beta_{2}+\beta_{3}+\beta_{6},$

$\mu_{\beta_{7}}=3\beta_{1}+2\beta_{2}+\beta_{3}+\beta_{7}$

$\Phi=E_{8}$

$\mu_{\beta_{1}}=\beta_{1},\leavevmode\nobreak\ \mu_{\beta_{2}}=\beta_{1}+\beta_{2},$

$\mu_{\beta_{3}}=2\beta_{1}+\beta_{2}+\beta_{3},\leavevmode\nobreak\ \mu_{\beta_{4}}=3\beta_{1}+2\beta_{2}+\beta_{3}+\beta_{4},$

$\mu_{\beta_{5}}=3\beta_{1}+\beta_{2}+\beta_{3}+\beta_{5},\leavevmode\nobreak\ \mu_{\beta_{6}}=3\beta_{1}+2\beta_{2}+\beta_{3}+\beta_{4}+\beta_{6},$

$\mu_{\beta_{7}}=4\beta_{1}+3\beta_{2}+\beta_{3}+\beta_{4}+\beta_{7},\leavevmode\nobreak\ \mu_{\beta_{8}}=5\beta_{1}+3\beta_{2}+2\beta_{3}+\beta_{4}+\beta_{8}$

$\Phi=F_{4}$

$\mu_{\beta_{1}}=\beta_{1},\leavevmode\nobreak\ \mu_{\beta_{2}}=\beta_{1}+\beta_{2},$

$\mu_{\beta_{3}}=2\beta_{1}+\beta_{2}+\beta_{3},\leavevmode\nobreak\ \mu_{\beta_{4}}=3\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4}$

$\Phi=G_{2}$ $\mu_{\beta_{1}}=\beta_{1},\leavevmode\nobreak\ \mu_{\beta_{2}}=\beta_{1}+\beta_{2}$

**Remark 2.2. **i) In fact, we will use Tables Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. only for exceptional root systems, but for the reader’s convenience we describe the weights $\mu_{\beta_{i}}$ for all irreducible root systems.

ii) Note that the correspondence between $\beta_{i}$ and $\mu_{\beta_{i}}$ is uniquely determined by the fact that

[TABLE]

where $\langle\cdot\rangle_{\mathbb{R}}$, as usual, denotes the linear span over $\mathbb{R}$. Furthermore, each $\beta\in\mathcal{B}$ occurs in $\mu_{\beta}$ with coefficient $1$. Note also that if $\beta\in\mathcal{B}\cap\Delta$ then $\mu_{\beta}$ is the unique weight in Table Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. in which expression $\beta$ occurs. Our enumeration of the weights $\mu_{\beta_{i}}$ slightly differs from [Pa2], [Ig1] and [IP].

iii) Recall that $\{e_{\alpha}^{*},\leavevmode\nobreak\ \alpha\in\Phi^{+}\}$ is the basis of $\mathfrak{n}^{*}$ dual to the basis $\{e_{\alpha},\leavevmode\nobreak\ \alpha\in\Phi^{+}\}$ of $\mathfrak{n}$. Put

[TABLE]

and denote by $X$ the union of all $N$-orbits in $\mathfrak{n}^{*}$ of elements of $R$. In fact, $X$ is a single $B$-orbit in $\mathfrak{n}^{*}$, and the $N$-orbits of two distinct point of $R$ are disjoint [Ko1, Theorem 2.5]. Kostant [Ko1, Theorems 2.8, 3.11] proved that $X$ is a Zariski dense subset of $\mathfrak{n}^{*}$, and for $t\in R$, up to scalar, for each $\beta_{i}\in\mathcal{B}$,

[TABLE]

(Here we identify $S(\mathfrak{n})$ with the algebra of polynomial functions on $\mathfrak{n}^{*}$.) Clearly, $r_{\beta_{i}}(\beta)$ is nothing but the coefficient at $\beta$ in the expression of $\mu_{\beta_{i}}$ in Table Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1..

We fix the generators $\xi_{\beta}$, $\beta\in\mathcal{B}$, so that the formulas (4) are satisfied (without any additional scalars). Recall the definition of the map $\sigma$ from (2). For $\beta\in\mathcal{B}$, we denote $\Delta_{\beta}=\sigma(\xi_{\beta})\in Z(\mathfrak{n})$. Explicit formulas for $\xi_{\beta}$ and $\Delta_{\beta}$ for classical root systems can be found in [IP, Subsection 2.1].

**Definition 2.3. **We call $\Delta_{\beta}$ (respectively, $\xi_{\beta}$), $\beta\in\mathcal{B}$, the canonical generators of the algebra $Z(\mathfrak{n})$ (respectively, of the algebra $Y(\mathfrak{n})$).

3. Centrally generated ideals

Let $\mathfrak{g}$, $\mathfrak{n}$, $\Phi$, $\Delta$, etc., be as in Section Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1.. A (two-sided) ideal $J\subset U(\mathfrak{n})$ is called primitive if $J$ is the annihilator of a simple $\mathfrak{n}$-module. An ideal $J$ is called centrally generated if $J$ is generated (as an ideal) by its intersection $J\cap Z(\mathfrak{n})$ with the center $Z(\mathfrak{n})$ of $U(\mathfrak{n})$.

In the 1960s Kirillov, Kostant and Souriau discovered that the orbits of the coadjoint action play a crucial role in the representation theory of $B$ and $N$ (see, e.g., [Ki1], [Ki2]). Works of Dixmier, Duflo, Vergne, Mathieu, Conze and Rentschler led to the result that the orbit method provides a nice description of primitive ideals of the universal enveloping algebra of a nilpotent Lie algebra (in particular, of $\mathfrak{n}$). Below we briefly recall this description.

To any linear form $\lambda\in\mathfrak{n}^{*}$ one can assign a bilinear form $\beta_{\lambda}$ on $\mathfrak{n}$ by putting $\beta_{\lambda}(x,y)=\lambda([x,y])$ for $x,y\in\mathfrak{n}$. A subalgebra $\mathfrak{p}\subseteq\mathfrak{n}$ is a polarization of $\mathfrak{n}$ at $\lambda$ if it is a maximal $\beta_{\lambda}$-isotropic subspace. By [Ve], such a subalgebra always exists. Let $\mathfrak{p}$ be a polarization of $\mathfrak{n}$ at $\lambda$, and let $W$ be the one-dimensional representation of $\mathfrak{p}$ defined by $x\mapsto\lambda(x)$. Then the annihilator $J(\lambda)=\mathrm{Ann}\,_{U(\mathfrak{n})}{V}$ of the induced representation $V=U(\mathfrak{n})\otimes_{U(\mathfrak{p})}W$ is a primitive two-sided ideal of $U(\mathfrak{n})$. It turns out that $J(\lambda)$ depends only on $\lambda$ and not on the choice of polarization. Further, $J(\lambda)=J(\mu)$ if and only if the coadjoint $N$-orbits of $\lambda$ and $\mu$ coincide. Finally, the Dixmier map

[TABLE]

induces a homeomorphism between $\mathfrak{n}^{*}/N$ and $\mathrm{Prim}\,U(\mathfrak{n})$, where the latter set is endowed with the Jacobson topology. (See [Di2], [Di4], [BGR] for the details.)

In addition, it is well known that the following conditions on an ideal $J\subset U(\mathfrak{n})$ are equivalent[Di4, Proposition 4.7.4, Theorem 4.7.9]:

[TABLE]

Recall that the Weyl algebra $\mathcal{A}_{s}$ of $2s$ variables is the unital associative algebra with generators $p_{i}$, $q_{i}$ for $1\leq i\leq s$, and relations $[p_{i},q_{i}]=1$, $[p_{i},q_{j}]=0$ for $i\neq j$, $[p_{i},p_{j}]=[q_{i},q_{j}]=0$ for all $i,\leavevmode\nobreak\ j$. Furthermore, in (5) we have $U(\mathfrak{n})/J\cong\mathcal{A}_{s}$ where $s$ equals one half of the dimension of the coadjoint $N$-orbit of $\lambda$, given that $J=J(\lambda)$.

**Definition 3.1. **To a map $\xi\colon\mathcal{B}\to\mathbb{C}$ we assign the linear form $f_{\xi}=\sum_{\beta\in\mathcal{B}}\xi(\beta)e_{\beta}^{*}\in\mathfrak{n}^{*}$. We call a form $f_{\xi}$ a Kostant form if $\xi(\beta)\neq 0$ for any $\beta\in\mathcal{B}\setminus\Delta$.

Let $V$ be a simple (and hence at most countably dimensional) $\mathfrak{n}$-module and let $J=\mathrm{Ann}\,_{U(\mathfrak{n})}{V}$ be the corresponding primitive ideal of $U(\mathfrak{n})$. By a version of Schur’s Lemma [Di1], each central element of $U(\mathfrak{n})$ acts on $V$ as a scalar operator. Given a tuple $c=(c_{\beta})_{\beta\in\mathcal{B}}$ of complex numbers, we denote by $J_{c}$ the ideal of $U(\mathfrak{n})$ generated by all $\Delta_{\beta}-c_{\beta}$, $\beta\in\mathcal{B}$. If, for each $\beta\in\mathcal{B}$, $\Delta_{\beta}$ acts on $V$ by the scalar $c_{\beta}$, then, clearly, $J_{c}\subseteq J$. Further, since $Z(\mathfrak{n})$ is a polynomial ring and the center of $U(\mathfrak{n})/J$ is trivial, $J$ is centrally generated if and only if $J=J_{c}$.

The following result was proved in [IP, Theorem 3.1] and [Ig1, Theorem 2.4].

Theorem 3.2. Suppose $\Phi$ is an irreducible root system of classical type, i.e., $\Phi=A_{n-1}$, $B_{n}$, $C_{n}$ or $D_{n}$. The following conditions on a primitive ideal $J\subset U(\mathfrak{n})$ are equivalent:

[TABLE]

If these conditions are satisfied, then the map $\xi$ can be reconstructed from $J$.

The main result of the paper is to prove that this is also true for exceptional root systems, see Theorem Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. in Section 4. One of the key ingredients in the proof of Theorem Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. was to check that if condition (ii) is satisfied then $J_{c}$ is primitive. To do this for $A_{n-1}$ and $C_{n}$, in [IP] an explicit set of generators of the quotient algebra $U(\mathfrak{n})/J_{c}$ was constructed. It turns out that these generators satisfy (up to scalars) the defining relations of the Weyl algebra $\mathcal{A}_{s}$ for $s=|\Phi^{+}\setminus\mathcal{B}|/2$. Since $\mathcal{A}_{s}$ is simple and, as one can check, $J_{c}\neq U(\mathfrak{n})$, we conclude that $U(\mathfrak{n})/J_{c}\cong\mathcal{A}_{s}$, and, consequently, $J_{c}$ is primitive. On the other hand, for $B_{n}$ and $D_{n}$, in [Ig1] an explicit set of generators for $U(\mathfrak{n})/J_{c}$ was constructed a posteriori (see [Ig1, Theorem 2.9]), while primitivity of $J_{c}$ was established by another argument. In this section we modify the idea from [Ig1, Proposition 2.5] to check that $J_{c}$ is primitive if $c_{\beta}\neq 0$ for $\beta\in\mathcal{B}\setminus\Delta$ for exceptional types.

To do this, we need some additional notation. From now to the end of this section we assume that $\Phi$ is an irreducible root system of exceptional type, i.e., $\Phi=E_{6}$, $E_{7}$, $E_{8}$, $F_{4}$ or $G_{2}$. Recall that $\beta_{1}$ is the maximal root with respect to the natural order on $\Phi$. It is obvious that $(\alpha,\beta_{1})\geq 0$ for all $\alpha\in\Phi^{+}$. We put $\widetilde{\Phi}=\{\alpha\in\Phi\mid(\alpha,\beta_{1})=0\}$ and $\widetilde{\Phi}^{+}=\widetilde{\Phi}\cap\Phi^{+}$, $\widetilde{\Delta}=\widetilde{\Phi}\cap\Delta$. Then $\widetilde{\Phi}$ is of respective type $D_{5}$, $E_{6}$, $E_{7}$, $C_{3}$ or $A_{1}$. Denote

[TABLE]

where $\langle\cdot\rangle_{\mathbb{C}}$ is the $\mathbb{C}$-linear span. Then $\widetilde{\mathfrak{n}}$ is a Lie subalgebra of $\mathfrak{n}$ isomorphic to the nilradical of the Borel subalgebra $\widetilde{\mathfrak{b}}=\widetilde{\mathfrak{g}}\cap\mathfrak{b}$ of the simple Lie algebra $\widetilde{\mathfrak{g}}$ with the root system $\widetilde{\Phi}$, where $\widetilde{\mathfrak{g}}$ is the subalgebra of $\mathfrak{g}$ generated by the root vectors $e_{\alpha}$, $\alpha\in\widetilde{\Phi}$.

On the other hand, $\mathfrak{k}$ is an ideal of $\mathfrak{n}$ isomorphic to the Heisenberg Lie algebra $\mathfrak{hei}_{s}$, where ${s=(|\Phi^{+}\setminus\widetilde{\Phi}^{+}|-1)/2}$, with the center $\mathbb{C}e_{\beta_{1}}$. (This follows from the fact that if $\alpha\in\Phi^{+}$ and $(\alpha,\beta_{1})>0$ then $\beta_{1}-\alpha$ is again a positive root, see [Jo2, Corollary 2.3] for the details.) Recall that $\mathfrak{hei}_{s}$ is the $(2s+1)$-dimensional Lie algebra with basis $\{z,\leavevmode\nobreak\ x_{i},\leavevmode\nobreak\ y_{i},\leavevmode\nobreak\ 1\leq i\leq s\}$ and relations $[x_{i},\leavevmode\nobreak\ y_{i}]=z$ for all $i$, $[x_{i},z]=[y_{j},z]=[x_{i},y_{j}]=0$ for all $i\neq j$.

Given $c_{1}\in\mathbb{C}^{\times}$, denote by $J_{1}$ the ideal of $U(\mathfrak{k})$ generated by $e_{\beta_{1}}-c_{1}$, so, clearly, $U(\mathfrak{k})/J_{1}\cong\mathcal{A}_{s}$. Since $\mathfrak{k}$ is an ideal of the Lie algebra $\mathfrak{n}$, given $x\in\widetilde{\mathfrak{n}}$, one can consider $\mathrm{ad}_{x}$ as a derivation of $\mathfrak{k}$. Since $e_{\beta_{1}}-c_{1}$ is a central element of $U(\mathfrak{n})$, one has $\mathrm{ad}_{x}(J_{1})\subseteq J_{1}$, so $\mathrm{ad}_{x}$ can be considered as a derivation of $\mathcal{A}_{s}$. It is well known (see, e.g., [Di4, 10.1.1–10.1.4]) that there exists unique $\theta(x)\in\mathcal{A}_{s}$ such that $\mathrm{ad}_{x}(y)=[\theta(x),y]$ for all $y\in\mathcal{A}_{s}$, and $\theta\colon\widetilde{\mathfrak{n}}\to\mathcal{A}_{s}$ is a morphism of Lie algebras. Furthermore, there exists the unique epimorphism of associative algebras $r\colon U(\mathfrak{n})\twoheadrightarrow U(\widetilde{\mathfrak{n}})\otimes\mathcal{A}_{s}$ such that $r(y)=1\otimes\overline{y}$ for $y\in\mathfrak{k}$ and $r(x)=x\otimes 1+1\otimes\theta(x)$ for $x\in\widetilde{\mathfrak{n}}$. (Here $\overline{a}$ is the image of an element $a\in U(\mathfrak{k})$ under the canonical projection $U(\mathfrak{k})\twoheadrightarrow U(\mathfrak{k})/J_{1}\cong\mathcal{A}_{s}$.) It turns out that the kernel of the epimorphism $r$ coincides with the ideal $J_{0}$ of $U(\mathfrak{n})$ generated by $e_{\beta_{1}}-c_{1}$ [Di4, Lemma 10.1.5].

Proposition 3.3. Let $J_{c}$ be the ideal of $U(\mathfrak{n})$ generated by $\Delta_{\beta}-c_{\beta}$ for all $\beta\in\mathcal{B}$ with $c_{\beta}\neq 0$ for $\beta\in\mathcal{B}\setminus\Delta$. Then $J_{c}$ is primitive.

Proof. Put $c_{1}=c_{\beta_{1}}$. Since $r$ is surjective, $r(J_{c})$ is an ideal of $U(\widetilde{\mathfrak{n}})\otimes\mathcal{A}_{s}$ generated by $r(\Delta_{\beta})-c_{\beta}$, $\beta\in\widetilde{\mathcal{B}}=\mathcal{B}\setminus\{\beta_{1}\}$. Note that $\widetilde{\mathcal{B}}$ is the Kostant cascade of $\widetilde{\Phi}$. Denote by $\widetilde{\Delta}_{\beta}$, $\beta\in\widetilde{\Phi}$, the set of canonical generators of $Z(\widetilde{\mathfrak{n}})$. We will show that, up to nonzero scalar, $r(\Delta_{\beta})$ coincides with $\widetilde{\Delta}_{\beta}\otimes 1$ for all $\beta\in\widetilde{\mathcal{B}}$.

To check this fact, we will use Tables Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1.. Pick a root $\beta\in\widetilde{\mathcal{B}}$. Since $r$ is surjective, $r(\Delta_{\beta})$ is central in $U(\widetilde{\mathfrak{n}})\otimes\mathcal{A}_{s}$. The center of this algebra has the form $Z(\widetilde{\mathfrak{n}})\otimes\mathbb{C}$, so in fact $r(\Delta_{\beta})\in Z(\widetilde{\mathfrak{n}})\otimes\mathbb{C}$. Denote $\widetilde{\mathfrak{h}}=\widetilde{\mathfrak{g}}\cap\mathfrak{h}$, so $\widetilde{\mathfrak{h}}$ is a Cartan subalgebra of $\widetilde{\mathfrak{g}}$ and $\widetilde{\mathfrak{b}}=\widetilde{\mathfrak{h}}\oplus\widetilde{\mathfrak{n}}$ as vector spaces. By [Ko2, Theorem 6] (see also [Jo1, Lemma 4.4]), $Z(\mathfrak{n})$ (respectively, $Z(\widetilde{\mathfrak{n}})$) is a direct sum of one-dimensional weight spaces of $\mathfrak{h}$ (respectively, of $\widetilde{\mathfrak{h}}$) with respect to the adjoint action of the corresponding Cartan subalgebras. Since $[h,e_{\beta_{1}}]=0$ for all $h\in\widetilde{\mathfrak{h}}$, the algebra $\widetilde{\mathfrak{h}}$ naturally acts on $\mathcal{A}_{s}$, and so on $U(\widetilde{\mathfrak{n}})\otimes\mathcal{A}_{s}$. We define the result of this action by $h.x$, $h\in\widetilde{\mathfrak{h}}$, $x\in U(\widetilde{\mathfrak{n}})\otimes\mathcal{A}_{s}$. Hence it is enough to check that, given $\beta\in\widetilde{\mathcal{B}}$, $r(\Delta_{\beta})$ is a nonzero $\widetilde{\mathfrak{h}}$-weight element of weight $\widetilde{\mu}_{\beta}=\mu_{\beta}-r_{\beta}(\beta_{1})\beta_{1}$, because $r_{\beta}(\beta_{1})$ is just the coefficient of $\beta_{1}$ in the expression of $\mu_{\beta}$ as a linear combination of roots from $\mathcal{B}$.

To prove that $r(\Delta_{\beta})$ is an $\widetilde{\mathfrak{h}}$-weight element of weight $\widetilde{\mu}_{\beta}$, denote the result of the natural (adjoint) action of $\widetilde{\mathfrak{h}}$ on $U(\mathfrak{n})$ by $h\cdot x$, $h\in\widetilde{\mathfrak{h}}$, $x\in U(\mathfrak{n})$. As above, since $\widetilde{\mathfrak{h}}\cdot e_{\beta_{1}}=0$, the algebra $\widetilde{\mathfrak{h}}$ naturally acts on $U(\mathfrak{n})/J_{0}$ by the formula $h\cdot r(x)=r(h\cdot x)$. We claim that this action coincides with the action of $\widetilde{\mathfrak{h}}$ on $U(\widetilde{\mathfrak{n}})\otimes\mathcal{A}_{s}$ defined above, i.e., $h.x=h\cdot x$ for all $h\in\widetilde{\mathfrak{h}}$, $x\in U(\widetilde{\mathfrak{n}})\otimes\mathcal{A}_{s}$.

Indeed, if $y\in\mathfrak{k}$, then

[TABLE]

as required. On the other hand, if $e_{\alpha}\in\widetilde{\mathfrak{n}}$ for some root $\alpha\in\Phi^{+}$, then, by [Jo1, Subsection 4.8], $\theta(e_{\alpha})$ is a linear combination of elements of the form $\overline{e}_{\alpha+\gamma}\overline{e}_{\beta_{1}-\gamma}$, $\gamma\in\Phi^{+}\setminus\widetilde{\Phi}^{+}$ (i.e., $(\gamma,\beta_{1})>0$). We conclude that

[TABLE]

because $\beta_{1}(\widetilde{\mathfrak{h}})=0$. Thus, we obtain

[TABLE]

It remains to note that $\Delta_{\beta}$ is an $\mathfrak{h}$-weight element of $U(\mathfrak{n})$ of weight $r_{\beta}(\beta_{1})\beta_{1}+\widetilde{\mu}_{\beta}$, but $\beta_{1}(\widetilde{\mathfrak{h}})=0$.

To show that $r(\Delta_{\beta})\neq 0$, recall that the kernel of $r$ is $J_{0}$. If $\Delta_{\beta}\in J_{0}$ (i.e., if $\Delta_{\beta}=(e_{\beta_{1}}-c_{\beta_{1}})a$ for some $a\in U(\mathfrak{n})$), then, clearly, $a\in Z(\mathfrak{n})$. But this contradicts the fact that $\Delta_{\beta}$ and $\Delta_{\beta_{1}}$ are algebraically independent, because, as one can deduce from [Pa2], $\Delta_{\beta_{1}}=e_{\beta_{1}}$ for all irreducible root systems.

So, given $\beta\in\widetilde{\mathcal{B}}$, there exists unique $a_{\beta}\in\mathbb{C}^{\times}$ such that $r(\Delta_{\beta})=a_{\beta}\widetilde{\Delta}_{\beta}\otimes 1$. Consequently, $r(J_{c})$ is generated by $\widetilde{\Delta}_{\beta}\otimes 1-\widetilde{c}_{\beta}$, $\beta\in\widetilde{\mathcal{B}}$, where $\widetilde{c}_{\beta}=a_{\beta}^{-1}c_{\beta}$. In particular, $\widetilde{c}_{\beta}\neq 0$ if $\beta\in\widetilde{\mathcal{B}}$ is not a simple root of $\widetilde{\Phi}^{+}$. Now we will use the induction on $\mathrm{rk}\,\Phi$ to prove that $J_{c}$ is primitive. The base (i.e., the case of classical $\Phi$ of low rank) immediately follows from [IP, Theorem 3.1] and [Ig1, Theorem 2.4]. Denote by $\widetilde{J}_{c}$ the ideal of $U(\widetilde{\mathfrak{n}})$ generated by $\widetilde{\Delta}_{\beta}-\widetilde{c}_{\beta}$, $\beta\in\widetilde{\mathcal{B}}$. By the inductive assumption, $\widetilde{J}_{c}$ is a primitive ideal of $U(\widetilde{\mathfrak{n}})$, so $U(\widetilde{\mathfrak{n}})/\widetilde{J}_{c}\cong\mathcal{A}_{t}$ for certain $t$. We conclude that

[TABLE]

Thus, $J_{c}$ is primitive. The proof is complete.$\square$

4. Distinct coadjoint orbits

Recall that, given a primitive ideal $J$ in $U(\mathfrak{n})$, there exist unique scalars $c_{\beta}\in\mathbb{C}$ such that $\Delta_{\beta}-c_{\beta}\in J$ for all $\beta\in\mathcal{B}$. To prove our main result, Theorem Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., we need to check that if $J$ is centrally generated then $c_{\beta}\neq 0$ for $\beta\in\mathcal{B}\setminus\Delta$. To do this, we will prove that certain coadjoint $N$-orbits on $\mathfrak{n}^{*}$ are distinct.

Namely, let $D$ be a subset of $\Phi^{+}$. To each map $\xi\colon D\to\mathbb{C}^{\times}$ one can assign the linear form

[TABLE]

Denote by $\Omega_{D,\xi}$ the coadjoint $N$-orbit of $f_{D,\xi}$. We say that $f_{D,\xi}$ and $\Omega_{D,\xi}$ are associated with the subset $D$. For example, $f_{D,\xi}$ is a Kostant form if and only if $\mathcal{B}\setminus\Delta\subset D\subset\mathcal{B}$.

It was proved in [Pa1, Corollary 1.4] that if $\Phi=A_{n-1}$, $D$ is an orthogonal subset (i.e., $(\alpha,\beta)=0$ for all $\alpha,\leavevmode\nobreak\ \beta\in D$, $\alpha\neq\beta$) and $\xi_{1}$, $\xi_{2}$ are two distinct maps from $D$ to $\mathbb{C}^{\times}$ then $\Omega_{D,\xi_{1}}\neq\Omega_{D,\xi_{2}}$. It is not hard to deduce from this result that the same is true for all classical root systems, see the proofsof [IP, Theorem 3.1] and [Ig1, Theorem 2.4]. But for exceptional types this is not an immediate consequence of the result for $A_{n-1}$. In this section, we prove that if $\xi_{1}\neq\xi_{2}$ then $\Omega_{D,\xi_{1}}$ and $\Omega_{D,\xi_{2}}$ are distinct for some particular orthogonal subsets $D$ and some particular maps $\xi_{1}$, $\xi_{2}$, which will be used in the next section in the proof of our main result.

To do this, we need to introduce the notion of singular roots.

**Definition 4.1. **Let $\beta$, $\alpha$ be positive roots. We say that $\alpha$ is $\beta$-singular (or singular for $\beta$) if there exists $\gamma\in\Phi^{+}$ such that $\beta=\alpha+\gamma$. The set of all $\beta$-singular roots is denoted by $S(\beta)$.

Note that if $\Phi$ is irreducible and simple-laced (i.e., if all roots in $\Phi$ have the same length) then, given $\beta>\alpha$, $\alpha$ is $\beta$-singular if and only if $(\alpha,\beta)>0$. It turns out that if $D$ is an orthogonal subset of $\Phi^{+}$, $\xi$ is a map from $D$ to $\mathbb{C}^{\times}$, and $\beta$, $\beta^{\prime}\in D$ are such that $\beta^{\prime}\in S(\beta)$ then $\Omega_{D,\xi}=\Omega_{D^{\prime},\xi^{\prime}}$, where $D^{\prime}=D\setminus\beta^{\prime}$ and $\xi^{\prime}$ is the restriction of $\xi$ to $D^{\prime}$ [Ig2, Lemma 1.3].

Proposition 4.2. Let $\Phi$ be an irreducible root system, and let $D$ be a subset of $\Phi^{+}$ such that if $\beta_{1}$, $\beta_{2}\in D$ then $\beta_{1}\notin S(\beta_{2})$. Let $\beta_{0}$ be a root in $D$, and let $\xi_{1}$, $\xi_{2}$ be maps from $D$ to $\mathbb{C}^{\times}$ for which $\xi_{1}(\beta_{0})\neq\xi_{2}(\beta_{0})$. Fix a total order $\leq_{t}$ on $\{e_{\alpha},\leavevmode\nobreak\ \alpha\in\Phi^{+}\}$ such that if $\alpha>\beta$ then $e_{\alpha}<_{t}e_{\beta}$. Assume that there exists a simple root $\alpha_{0}\in\Delta$ satisfying $(\alpha_{0},\beta_{0})\neq 0$ and $(\alpha_{0},\beta)=0$ for all $\beta\in D$ such that $e_{\beta}<_{t}e_{\beta_{0}}$. Then $\Omega_{D,\xi_{1}}\neq\Omega_{D,\xi_{2}}$.

Proof. As usual, given a vector space $V$, we denote by $\mathfrak{gl}(V)$ the Lie algebra of all linear operators on $V$. Recall that $\Phi$ is the root system of the Lie algebra $\mathfrak{g}$. Denote by $\mathrm{ad}\colon\mathfrak{g}\to\mathfrak{gl}(\mathfrak{g})$ the adjoint representation of the Lie algebra $\mathfrak{g}$, i.e., $\mathrm{ad}(x)=\mathrm{ad}_{x}$. It is well known that the adjoint representation is faithful, so $\mathrm{ad}(\mathfrak{g})$ and $\mathfrak{g}$ are isomorphic as Lie algebras. Let $\mathrm{rk}\,\Phi=n$. To each simple root $\alpha_{i}$, $1\leq i\leq n$, one can assign unique $h_{\alpha_{i}}\in\mathfrak{h}$ such that $\alpha_{i}(h_{\alpha_{j}})=2(\alpha_{i},\alpha_{j})/(\alpha_{j},\alpha_{j})$ for all $i,\leavevmode\nobreak\ j$. Recall that $\{e_{\alpha},\leavevmode\nobreak\ \alpha\in\Phi^{+}\}$ is a basis of $\mathfrak{n}$. It can be uniquely extended to the Chevalley basis $\{e_{\alpha},\leavevmode\nobreak\ \alpha\in\Phi^{+}\}\cup\{h_{\alpha_{i}},\leavevmode\nobreak\ 1\leq i\leq n\}\cup\{e_{-\alpha},\leavevmode\nobreak\ \alpha\in\Phi^{+}\}$ of $\mathfrak{g}$. We extend the total order $\leq_{t}$ on the set $\{e_{\alpha}.\leavevmode\nobreak\ \alpha\in\Phi^{+}+\}$ to a total order $\leq_{t}$ on this basis such that $e_{\alpha}<_{t}h_{\alpha_{i}}<_{t}e_{-\beta}$ for all $\alpha,\beta\in\Phi^{+}$, $1\leq i\leq n$, and $e_{\alpha}<_{t}e_{\beta}$ if $\alpha,\beta\in\Phi$ and $\alpha>\beta$. This identifies $\mathfrak{gl}(\mathfrak{g})$ with the Lie algebra $\mathfrak{gl}_{\dim\mathfrak{g}}(\mathbb{C})$, and $\mathrm{ad}(\mathfrak{n})$ with a subalgebra of the Lie algebra $\mathfrak{u}$ of all upper-triangular matrices from $\mathfrak{gl}_{\dim\mathfrak{g}}(\mathbb{C})$ with zeroes on the diagonal.

Let $\mathrm{GL}(V)$ be the group of all invertible linear operators on a vector space $V$. Since we fixed a basis in $\mathfrak{g}$, the group $\mathrm{GL}(\mathfrak{g})$ is identified with the group $\mathrm{GL}_{\dim\mathfrak{g}}(\mathbb{C})$, and $\exp\mathrm{ad}(\mathfrak{n})\cong N$ is identified with a subgroup of the group $U$ of all upper-triangular matrices from $\mathrm{GL}_{\dim\mathfrak{g}}(\mathbb{C})$ with $1$’s on the diagonal. Furthermore, using the Killing form on $\mathfrak{g}$ and the trace form on $\mathfrak{gl}(\mathfrak{g})$, one can identify $\mathfrak{n}^{*}$ with the space $\mathfrak{n}_{-}=\langle e_{-\alpha},\leavevmode\nobreak\ \alpha\in\Phi^{+}\rangle_{\mathbb{C}}$ and $\mathfrak{u}^{*}$ with the space $\mathfrak{u}_{-}=\mathfrak{u}^{T}$, where the superscript $T$ denote the transposed matrix. Under these identifications, it is enough to check that the coadjoint $U$-orbits of the linear forms $\widetilde{f}_{D,\xi_{1}}$ and $\widetilde{f}_{D,\xi_{2}}$ are distinct. Here, given $\xi\colon D\to\mathbb{C}^{\times}$, we denote by $\widetilde{f}_{D,\xi}$ the matrix

[TABLE]

To do this, we will study the matrix $f=\widetilde{f}_{D,\xi}$ in more detail. The rows and the columns of matrices from $\mathfrak{gl}(\mathfrak{g})$ are now indexed by the elements of the Chevalley basis fixed above. Given a matrix $x$ from $\mathfrak{gl}(\mathfrak{g})$ and basis elements $a,\leavevmode\nobreak\ b$, we will denote by $x_{a,b}$ the entry of $x$ lying in the $a$th row and the $b$th column. Since

[TABLE]

we obtain $f_{h_{\alpha_{0}},e_{\beta_{0}}}=-\xi(\beta_{0})\dfrac{2(\alpha_{0},\beta_{0})}{(\alpha_{0},\alpha_{0})}\neq 0$. One may assume without loss of generality that $h_{\alpha_{0}}>_{t}h_{\alpha_{i}}$ for all $\alpha_{i}\neq\alpha_{0}$. We claim that

[TABLE]

Indeed, if $\alpha\notin D$ then, evidently, $f_{h_{\alpha_{0}},e_{\alpha}}=0$. If $\alpha=\beta\in D$ and $e_{\beta}<_{t}e_{\beta_{0}}$ then $(\alpha_{0},\leavevmode\nobreak\ \beta)=0$, hence

[TABLE]

because $(\alpha_{0},\beta)=0$. On the other hand, if $f_{e_{-\gamma},e_{\beta_{0}}}\neq 0$ for some $\gamma\in\Phi^{+}$ then $\beta_{0}=\beta-\gamma$. This contradicts the condition $\beta_{0}\notin S(\beta)$.

Thus, $(\widetilde{f}_{D,\xi_{1}})_{h_{\alpha_{0}},e_{\alpha}}$ and $(\widetilde{f}_{D,\xi_{2}})_{h_{\alpha_{0}},e_{\alpha}}$ are different nonzero scalars, and (6) is satisfied both for $f=\widetilde{f}_{D,\xi_{1}}$ and for $f=\widetilde{f}_{D,\xi_{2}}$. Now it follows immediately from the proof of [An, Proposition 3] that the coadjoint $U$-orbits of these matrices are distinct, and, consequently, $\Omega_{D,\xi_{1}}\neq\Omega_{D,\xi_{2}}$, as required.$\square$

Now, for exceptional $\Phi$, we will prove an existence of certain subsets $D\subset\Phi^{+}$. To each such subset $D$ from this list we assign its subset $D^{\prime}\subset D$. Using Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., we will show that if $\xi_{1}$ and $\xi_{2}$ are two maps from $D$ to $\mathbb{C}^{\times}$ such that $\xi_{1}(\beta_{0})\neq\xi_{2}(\beta_{0})$ for some root $\beta_{0}\in D^{\prime}$ then $\Omega_{D,\xi_{1}}\neq\Omega_{D,\xi_{2}}$, see Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. below. We will consider all exceptional root systems subsequently. For brevity, we use the following notation. If $\beta=\sum_{i=1}^{n}m_{i}\alpha_{i}\in D$ then we write $m_{1}\ldots m_{n}$ instead of $\beta$ (our enumeration of simple roots is as in (1)). Note that in all cases, except 11, 12 and 14 for $F_{4}$ (see Table Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. in the proof of Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. below), $D$ is an orthogonal subset of $\Phi$, while in cases 11, 12, 14 for $F_{4}$ all inner products of distinct roots from $D$ are non-positive. Note also that all these subsets are linearly independent.

Proposition 4.3. Let $\Phi$ be an irreducible root system of exceptional type. Denote by $c$ an $m$-tuple $c=(c_{\beta_{1}},\leavevmode\nobreak\ \ldots,\leavevmode\nobreak\ c_{\beta_{m}})$, $m=|\mathcal{B}|$, and assume that $c_{\beta_{i}}=0$ for some $\beta_{i}\in\mathcal{B}\setminus\Delta$. There exist subsets $D,\leavevmode\nobreak\ D^{\prime}\subset\Phi^{+}$ such that $D$ is linearly independent, $D^{\prime}\subset D$, and the following conditions are satisfied:

[TABLE]

Furthermore, let $\mu_{\beta_{i}}=\sum\nolimits_{\gamma\in D}a_{\gamma,\beta_{i}}\gamma,\leavevmode\nobreak\ 1\leq i\leq m,\leavevmode\nobreak\ c_{\beta_{i}}\neq 0$. Then, for $\beta_{0}\in D$, the solution space of the system of linear equations $\sum\nolimits_{\gamma\in D}a_{\gamma,\beta_{i}}y_{\gamma}=b_{i},\leavevmode\nobreak\ 1\leq i\leq m,\leavevmode\nobreak\ c_{\beta_{i}}\neq 0,$ is not orthogonal to the axis $y_{\beta_{0}}$ if and only if $\beta_{0}\in D^{\prime}$.

Proof. The proof is case-by-case. Below we present the list of the subsets $D$, $D^{\prime}$ and the correspondence between these subsets and $m$-tuples $c$ for $E_{6}$, $F_{4}$ and $G_{2}$.$\square$

Table 4. List of the subsets $D$ and $D^{\prime}$ for $E_{6}$

Table 5. List of the subsets $D$ and $D^{\prime}$ for $F_{4}$

Table 6. Correspondence between $c$ and $D$ for $E_{6}$ and $F_{4}$

Table 7. List of the subsets $D$ and $D^{\prime}$ and correspondence between $c$ and $D$ for $G_{2}$

The root systems $E_{7}$, $E_{8}$ can be considered similarly. We list the subsets $D$, $D^{\prime}$ and prove the correspondence between $c$ and $D$ for $E_{7}$ and $E_{8}$ in Appendix A. (In fact, we just checked all orthogonal subsets of $\Phi^{+}$ to find subsets satisfying the required conditions using Python Programming Language [Py]. The listing of the code is presented in Appendix B.)

Proposition 4.4. Let $\Phi$ be an irreducible root system of exceptional type, and let $D\subset\Phi^{+}$ be a subset from Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1.. Let $\xi_{1}$, $\xi_{2}$ be maps from $D$ to $\mathbb{C}^{\times}$. There exists a root $\beta_{0}\in D^{\prime}$ such that if $\xi_{1}(\beta_{0})\neq\xi_{2}(\beta_{0})$ then $\Omega_{D,\xi_{1}}\neq\Omega_{D,\xi_{2}}$.

Proof. Note that if $\beta_{1},\leavevmode\nobreak\ \beta_{2}\in D$ then $\beta_{1}\notin S(\beta_{2})$. Indeed, for $E_{6}$, $E_{7}$ and $E_{8}$ this follows from the fact that $(\beta_{1},\leavevmode\nobreak\ \beta_{2})\leq 0$ for all distinct roots $\beta_{1}$, $\beta_{2}\in D$ and the comment before Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1.. For $F_{4}$ and $G_{2}$, this can be checked directly. According to Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., it is enough to check that there exist a simple root $\alpha_{0}$ and a root $\beta_{0}\in D^{\prime}$ such that $(\alpha_{0},\beta_{0})\neq 0$ and $(\alpha_{0},\beta)=0$ for all $\beta\in D\setminus D^{\prime}$. (Indeed, one can pick $\beta_{0}$ such that $e_{\beta_{0}}$ is minimal with respect to the total order $\leq_{t}$ from Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. among all $e_{\beta}$, $\beta\in D^{\prime}$, for which $\beta$ is not orthogonal to $\alpha_{0}$.) The existence of such roots follows from Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. (iv). For example, consider case 10 for $\Phi=E_{6}$. Here

[TABLE]

Then $\beta_{0}=\alpha_{3}+\alpha_{4}$ and $\alpha_{0}=\alpha_{4}$ satisfy all the conditions of Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., hence $\Omega_{D,\xi_{1}}\neq\Omega_{D,\xi_{2}}$.$\square$

5. Proof of the main result

We are now ready to formulate and, using the previous sections, prove our main result, Theorem Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. (cf. Theorem Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1.). Note that each element of $S(\mathfrak{n})$ can be considered as a polynomial function on $\mathfrak{n}^{*}$ via the natural isomorphism $(\mathfrak{n}^{*})^{*}\cong\mathfrak{n}$.

Theorem 5.1. Suppose $\Phi$ is an irreducible root system of exceptional type, i.e., $\Phi=E_{6}$, $E_{7}$, $E_{8}$, $F_{4}$ or $G_{2}$. The following conditions on a primitive ideal $J\subset U(\mathfrak{n})$ are equivalent:

[TABLE]

If these conditions are satisfied, then the map $\xi$ can be reconstructed from $J$.

Proof. $\mathrm{(ii)}\Longrightarrow\mathrm{(iii)}$. Recall that two $N$-orbits of distinct linear form from

[TABLE]

are disjoint, and the union $X$ of all such orbits is an open dense subset of $\mathfrak{n}^{*}$; in fact, $X$ is a single $B$-orbit. Given $\beta_{i}\in\mathcal{B}$ and $t\in R$,

[TABLE]

see Remark Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. (iii). Furthermore, from Remark Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. (i), (ii) we see that actually

[TABLE]

We claim that there exists a unique map $\xi\colon\mathcal{B}\to\mathbb{C}^{\times}$ such that $\xi(\beta)\neq 0$ for $\beta\in\mathcal{B}\setminus\Delta$ and the Kostant form $f_{\xi}$ satisfies $\xi_{\beta_{i}}(f_{\xi})=c_{\beta_{i}}$ for all $\beta_{i}\in\mathcal{B}$. Indeed, since

[TABLE]

belongs to $\overline{R}$, the Zariski closure of $R$ in $\mathfrak{n}^{*}$, we obtain that

[TABLE]

for all $i$ from $1$ to $m$. Since $\xi_{\beta_{1}}=e_{\beta_{1}}$, we must set $\xi(\beta_{1})=c_{\beta_{1}}$. Now, assume that $i>1$ and that $\xi_{\beta_{j}}$ is already defined for all $j<i$ so that $\xi_{\beta_{j}}(f_{\xi})=c_{\beta_{j}}$. Then one can put $\xi(\beta_{i})$ to be equal to $c_{\beta_{i}}\prod_{j<i}\xi(\beta_{j})^{-r_{\beta_{i}}(\beta_{j})}$, so that $\xi_{\beta_{i}}(f_{\xi})=c_{\beta_{i}}$, as required. Note that, according to Remark Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. (i), if $\xi(\beta_{j})=0$ for some $j<i$ then $\beta_{j}\in\mathcal{B}\cap\Delta$, and, consequently, $r_{\beta_{i}}(\beta_{j})=0$. Thus, $\beta_{j}$ does not actually occurs in the expression of $\mu_{\beta_{i}}$ and we do not divide by zero in the definition of $\xi(\beta_{i})$.

Now, let $\xi\colon\mathcal{B}\to\mathbb{C}^{\times}$ be such that $\xi_{\beta_{i}}(f_{\xi})=c_{\beta_{i}}$ for all $i$. Then, by [Di4, 6.6.9 (c)], $J(f_{\xi})$ contains $\Delta_{\beta_{i}}-c_{\beta_{i}}$ for all $i$, hence $J(f_{\xi})$ contains the centrally generated ideal $J_{c}$, which is generated by definition by all $\Delta_{\beta_{i}}-c_{\beta_{i}}$, $1\leq i\leq m$. But, thanks to Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., $J_{c}$ is primitive. Thus, both $J(f_{\xi})$ and $J_{c}$ are primitive (and so maximal), hence $J(f_{\xi})=J_{c}$.

$\mathrm{(iii)}\Longrightarrow\mathrm{(i)}$. Again by [Di4, 6.6.9 (c)],

[TABLE]

for all $\beta_{i}\in\mathcal{B}$, hence $c_{\beta_{i}}\neq 0$ for $\beta_{i}\in\mathcal{B}\setminus\Delta$. The ideal $J=J(f_{\xi})$ contains the centrally generated ideal $J_{c}$, and condition (ii) is satisfied, so $J_{c}$ is primitive and $J=J(f_{\xi})=J_{c}$, as required.

$\mathrm{(i)}\Longrightarrow\mathrm{(ii)}$. This is the most interesting and complicated part of the proof. Let $J$ be a centrally generated primitive ideal of $U(\mathfrak{n})$. Denote by $c$ the $m$-tuple $c=(c_{\beta_{1}},\leavevmode\nobreak\ \ldots,\leavevmode\nobreak\ c_{\beta_{m}})$ and assume that $c_{\beta_{i}}=0$ for some $\beta_{i}\in\mathcal{B}\setminus\Delta$. To such an $m$-tuple we assign subsets $D^{\prime}\subset D\subset\Phi^{+}$ from Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1..

Our next claim is that there exists a map $\eta$ from $D$ to the field of rational functions $\mathbb{C}(x)$ satisfying the following conditions (here, given $z\in\mathbb{C}^{\times}$, we denote by $\eta(z)$ the map from $D$ to $\mathbb{C}^{\times}$ defined by $D\ni\gamma\mapsto\eta(\gamma)(z)$):

[TABLE]

To construct such a map $\eta$, we first note that if $c_{\beta_{i}}\neq 0$ then $\mu_{\beta_{i}}$ can be uniquely expressed as a $\mathbb{Z}_{\geq 0}$-linear combination of the roots from $D$, i.e., there exist unique $a_{\gamma,\beta_{i}}\in\mathbb{Z}_{\geq 0}$, $\gamma\in D$, satisfying

[TABLE]

Indeed, the uniqueness follows from the linear independence of $D$ (Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. (i)), while the existence is implied by Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. (iii). For instance, if the root system $\Phi$ is of type $E_{6}$ and

[TABLE]

is the $8$th subset from Table Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. then

[TABLE]

This means that, given a map $\xi\colon D\to\mathbb{C}^{\times}$, the value of at most one monomial of $\xi_{\beta_{i}}$ at the linear form $f_{D,\xi}$ is non-zero (precisely, the monomial $e_{D}=\prod_{\gamma\in D}e_{\gamma}^{a_{\gamma,\beta_{i}}}$). Indeed, the value of $f_{D,\xi}$ on a monomial is nonzero if and only if all variables involved in this monomial has the form $e_{\gamma}$ for $\gamma\in D$, but all monomials of $\xi_{\beta_{i}}$ have weight $\mu_{\beta_{i}}$. (We see that if $c_{\beta_{j}}=0$ then $\xi_{\beta_{j}}(f_{D,\xi})=0$ by Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. (ii).) Note, however, that a priori we do not know that the monomial $e_{D}$ in fact occurs in $\xi_{\beta_{i}}$.

To check that $e_{D}$ occurs in $\xi_{\beta_{i}}$, we consider the subset

[TABLE]

(Actually, $\beta_{1}\in D$ if and only if $c_{\beta_{1}}\neq 0$.) Given a map $\xi\colon D\to\mathbb{C}^{\times}$, we consider a map $\xi_{1}\colon D_{1}\to\mathbb{C}^{\times}$ such that $\xi_{1}(\gamma)=\xi(\gamma)$ for $\gamma\in D$, and the linear form $f_{1}$, where

[TABLE]

Clearly, $e_{D}(f_{D,\xi})=e_{D}(f_{1})$. On the other hand, one can easily check that the condition $f_{1}(e_{\beta_{1}})\neq 0$ implies that there exists a linear form $\lambda$ in the coadjoint $N$-orbit of $f_{1}$ such that $\lambda(e_{\beta})\neq 0$ for all $\beta\in\mathcal{B}$. By Remark Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. (iii), $e_{\mathcal{B}}=\prod_{\beta\in\mathcal{B}}e_{\beta}^{r_{\beta_{i}}(\beta)}$ enters $\xi_{\beta_{i}}$ with coefficient 1, and $e_{\mathcal{B}}(\lambda)\neq 0$. But $\lambda=g.f_{1}$ for a certain $g\in N$, where $g.f_{1}$ denotes the result of the coadjoint action of $N$ on $\mathfrak{n}^{*}$. The adjoint action of $N$ on the algebra $S(\mathfrak{n})$ has the form

[TABLE]

We see that $(g^{-1}.e_{\mathcal{B}})(f_{1})=e_{\mathcal{B}}(\lambda)\neq 0$. But $\xi_{\beta_{i}}$ in $N$-invariant, so the monomial $g^{-1}.e_{\mathcal{B}}$ occurs in $\xi_{\beta}$ (with coefficient 1). Thus, the latter monomial coincides with $c_{D}^{i}e_{D}$ for certain $c_{D}^{i}\in\mathbb{C}^{\times}$.

Next, we note that the (affine) solution space for the system of linear equations

[TABLE]

is at least one-dimensional for all possible $b_{i}\in\mathbb{C}$. Indeed, $k=|D|$ is in fact not less than the number of non-zero scalars in $c$ plus one, so this system on $k$ variables contains at most $(k-1)$ equations. It follows from Panov’s description of the weights $\mu_{\beta_{i}}$ [Pa2, p. 8] that the equations are linearly independent, and the rank of the system is at most $(k-1)$. Hence the solution space is at least one-dimensional, as required. Recall that $D^{\prime}$ is exactly the set of roots $\gamma\in D$ for which the solution space is not orthogonal to the axis $y_{\gamma}$.

We are ready to construct a map $\eta$ satisfying the above conditions. Assume that such a map is already constructed. Then, given $z\in\mathbb{C}^{\times}$,

[TABLE]

Let $b_{i}=c_{\beta_{i}}-c_{D}^{i}$, and $y=(y_{\gamma})_{\gamma\in D}$ be a solution of system (8). Then, clearly, $\prod_{\gamma\in D}x_{\gamma}^{a_{\gamma,\beta_{i}}}=c_{\beta_{i}}\left(c_{D}^{i}\right)^{-1}$ for all $i$, where $x_{\gamma}=\exp(y_{\gamma})$. Let $\{y_{\gamma}(t)\}_{\gamma\in D}$ be a one-dimensional affine subspace of the space of solutions of (8) not orthogonal to the axes $y_{\gamma}$ for $\gamma\in D^{\prime}$. Note that each $y_{\gamma}(t)$, $\gamma\in D$, is an affine-linear function in $t$. We set $x=\exp(t)$ and $\eta(\gamma)(x)=\exp(y_{\gamma}(t))$ for $\gamma\in D$, then $\eta(\gamma)$ is a rational function in $x$. Condition (iii) is satisfied by the definition of $\eta$, condition (ii) is obvious. Finally, condition (i) is satisfied because the subspace $\{y_{\gamma}(t)\}_{\gamma\in D}$ is not orthogonal to the axis $y_{\gamma}$ for $\gamma\in D^{\prime}$.

For example, if $\Phi=E_{6}$ and $D=\{\gamma_{1},\leavevmode\nobreak\ \gamma_{2},\leavevmode\nobreak\ \gamma_{3},\leavevmode\nobreak\ \gamma_{4}\}$ is, as above, the 8th subset from Table Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., then one can put

[TABLE]

(Here, given $a\in\mathbb{C}$, we denote by $a^{1/2}$ a complex number such that $(a^{1/2})^{2}=a$.) Another example: if $\Phi=E_{6}$ and $D=\{\gamma_{1}=\alpha_{3}+\alpha_{4}+\alpha_{5}+\alpha_{6},\leavevmode\nobreak\ \gamma_{2}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+3\alpha_{4}+2\alpha_{5}+\alpha_{6},\leavevmode\nobreak\ \gamma_{3}=\alpha_{1}+\alpha_{3}+\alpha_{4}+\alpha_{5}\}$ is the 11th subset from Table Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., then $D^{\prime}=\{\gamma_{1},\leavevmode\nobreak\ \gamma_{3}\}$ and

[TABLE]

In general, let $\eta\colon D\to\mathbb{C}(x)$ be a map satisfying conditions (i), (ii), (iii). Since $\xi_{\beta_{i}}(f_{D,\eta(z)})=c_{\beta_{i}}$ for all $i$ and all $z\in\mathbb{C}^{\times}$, one has $J\subset J(f_{D,\eta(z)})$ for all $z\in\mathbb{C}^{\times}$. But both $J$ and $J(f_{D,\eta(z)})$ are maximal, hence $J=J(f_{D,\eta(z)})$ for all $z\in\mathbb{C}^{\times}$. Pick a root $\beta_{0}\in D^{\prime}$ satisfying Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1.. Since $\eta(\beta_{0})$ is non-constant, there exist $z_{1},\leavevmode\nobreak\ z_{2}\in\mathbb{C}^{\times}$ such that $\eta(\beta_{0})(z_{1})\neq\eta(\beta_{0})(z_{2})$. By Proposition Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., the coadjoint orbits of $f_{D,\eta(z_{1})}$ and $f_{D,\eta(z_{2})}$ do not coincide. Hence $J(f_{D,\eta(z_{1})})\neq J(f_{D,\eta(z_{2})})$, a contradiction.

Finally, if conditions (i)–(iii) are satisfied then formula (7) and the equality $\Delta_{\beta_{1}}=e_{\beta_{1}}$ together imply that the map $\xi$ can be reconstructed from $J$. The proof is complete.$\square$

As an immediate corollary, we obtain that a similar result is true for all (probably, reducible) root systems.

Theorem 5.2. Let $\Phi$ be an arbitrary root system. The following conditions on a primitive ideal $J\subset U(\mathfrak{n})$ are equivalent:

[TABLE]

If these conditions are satisfied, then the map $\xi$ can be reconstructed from $J$.

Proof. Let $\Phi_{i}$, $1\leq i\leq k$, be the irreducible components of the root system $\Phi$, and let $\mathfrak{n}=\bigoplus_{i=1}^{k}\mathfrak{n}_{k}$ be the corresponding division of $\mathfrak{n}$ into a direct sum of its nilpotent ideals, so $U(\mathfrak{n})=U(\mathfrak{n}_{1})\otimes\ldots\otimes U(\mathfrak{n}_{k})$ as associative algebras.

$\mathrm{(ii)}\Longrightarrow\mathrm{(iii)}$. Denote $\Delta_{i}=\Delta\cap\Phi_{i}$ and $\mathcal{B}_{i}=\mathcal{B}\cap\Phi_{i}$ for $1\leq i\leq k$, so $\Delta_{i}$ is a basis for $\Phi_{i}$ such that $\Phi_{i}^{+}=\Phi^{+}\cap\Phi_{i}$, and $\mathcal{B}_{i}$ is the Kostant cascade in $\Phi_{i}^{+}$. Put also $J_{i}=J\cap U(\mathfrak{n}_{i})$, so $J_{i}$ is an ideal of $U(\mathfrak{n}_{i})$ containing all $\Delta_{\beta}-c_{\beta}$, $\beta\in\mathcal{B}_{i}$. Since $c_{\beta}\neq 0$ for $\beta\in\mathcal{B}_{i}\setminus\Delta_{i}$, Theorem Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., [IP, Theorem 3.1] and [Ig1, Theorem 2.4] imply that $J_{i}$ is a primitive ideal of $U(\mathfrak{n}_{i})$. Furthermore, for each $i$, there exists a map $\xi_{i}\colon\mathcal{B}_{i}\to\mathbb{C}$ such that $f_{\xi_{i}}$ is a Kostant form on $\mathfrak{n}_{i}$ and $J_{i}=J(f_{\xi_{i}})$. Define $\xi$ to be the map from $\mathcal{B}$ to $\mathbb{C}$ such that $\xi(\beta)=\xi_{i}(\beta)$ for $\beta\in\mathcal{B}_{i}$, so $f_{\xi}$ is a Kostant form on $\mathfrak{n}$. Now, let $J_{c}$ be, as above, the ideal of $U(\mathfrak{n})$ generated by $\Delta_{\beta}-c_{\beta}$, $\beta\in\mathcal{B}$. Then $J_{c}$ is contained both in $J$ and in $J(f_{\xi})$. But the quotient algebra $U(\mathfrak{n}_{i})/J_{i}$ is isomorphic to the Weyl algebra $\mathcal{A}_{s_{i}}$ for certain $s_{i}\geq 1$. Thus,

[TABLE]

where $s=s_{1}+\ldots+s_{k}$, because $\mathcal{A}_{a}\otimes\mathcal{A}_{b}\cong\mathcal{A}_{a+b}$. It follows that $J_{c}$ is primitive (and so maximal), hence $J=J_{c}=J(f_{\xi})$.

$\mathrm{(iii)}\Longrightarrow\mathrm{(i)}$. Denote by $\xi_{i}$ the restriction of $\xi$ to $\mathcal{B}_{i}$, $1\leq i\leq k$. Again by Theorem Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1.,[IP, Theorem 3.1] and [Ig1, Theorem 2.4], each $J(f_{\xi_{i}})$ is a centrally generated primitive ideal of $U(\mathfrak{n}_{i})$, and $c_{\beta}\neq 0$ for all $\beta\in\mathcal{B}_{i}\setminus\Delta_{i}$ (and so for all $\beta\in\mathcal{B}\setminus\Delta$). And we see again that the centrally generated ideal $J_{c}$ is primitive and at the same time is contained in $J=J(f_{\xi})$, thus, $J=J_{c}$.

$\mathrm{(i)}\Longrightarrow\mathrm{(ii)}$. Assume that there exists $\beta\in\mathcal{B}\setminus\Delta$ such that $c_{\beta}=0$. Let $A$ be the set of all indices $a$ between 1 and $k$ such that there exists $\beta\in\mathcal{B}_{a}\setminus\Delta_{a}$ for which $c_{\beta}=0$. It follows from the proofs of Theorem Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., [IP, Theorem 3.1] and [Ig1, Theorem 2.4] that, given $a\in A$, there exist $D_{a}\subset\Phi_{j}^{+}$ and distinct maps $\xi_{a}^{1}$, $\xi_{a}^{2}$ from $D_{a}$ to $\mathbb{C}^{\times}$ such that

[TABLE]

for all $\beta\in\mathcal{B}_{a}$, and the coadjoint orbits of $f_{D_{a},\xi_{a}^{1}}$ and $f_{D_{a},\xi_{a}^{2}}$ (in $\mathfrak{n}_{a}^{*}$) are distinct.

One the other hand, if $i\in\{1,\leavevmode\nobreak\ \ldots,\leavevmode\nobreak\ k\}\setminus A$ then $c_{\beta}\neq 0$ for all $\beta\in\mathcal{B}_{i}\setminus\Delta_{i}$. Hence, as above, $J\cap U(\mathfrak{n}_{i})=J(f_{\xi_{i}})$ for a certain map $\xi_{i}\colon D\to\mathbb{C}$ (in other words, $f_{\xi_{i}}$ is a Kostant form on $\mathfrak{n}_{i}$ and $\xi_{\beta}(f_{\xi_{i}})=c_{\beta}$ for all $\beta\in\mathcal{B}_{i}$). Clearly, if $f\in\mathfrak{n}^{*}$ is a linear form on $\mathfrak{n}$ and $f_{j}$ is its restriction to $\mathfrak{n}_{j}$, $1\leq j\leq k$, then

[TABLE]

where $\Omega_{f}$ (respectively, $\Omega_{f_{j}}$) is the coadjoint orbit of the linear form $f$ in $\mathfrak{n}^{*}$ (respectively, of the linear form $f_{j}$ in $\mathfrak{n}_{j}^{*}$). Now, put

[TABLE]

and define $\xi^{j}\colon D\to\mathbb{C}$, $j=1,2$, by the rule

[TABLE]

Put $f^{j}=f_{D,\xi^{j}}$ for $j=1,\leavevmode\nobreak\ 2$, so

[TABLE]

where $\Omega_{\xi_{i}}$ is the coadjoint orbit of the Kostant form $f_{\xi_{i}}$ in $\mathfrak{n}_{i}^{*}$. Since $\Omega_{D_{a},\xi_{a}^{1}}\neq\Omega_{D_{a},\xi_{a}^{2}}$ for at least one $a\in A$, one has $\Omega_{f^{1}}\neq\Omega_{f^{2}}$, so $J(f^{1})\neq J(f^{2})$. At the same time, both maximal ideals contain the maximal ideal $J$, a contradiction.

Finally, if conditions (i)–(iii) are satisfied then the map $\xi$ can be reconstructed from $J$, because the restriction of $\xi$ to $\mathcal{B}_{i}$ can be reconstructed from $J_{i}$ for all $i$. The proof is complete.$\square$

Supplementary material. The online version of this article contains additional supplementary material (analogues of Tables Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1., Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types00footnotetext: The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘‘BASIS’’, grant no. 18–1–7–2–1. for the root systems $E_{7}$, $E_{8}$ and the listing of the code generating these analogues). Please visit https://arxiv.org/pdf/1907.04219.pdf.