This paper investigates special unipotent representations linked to complex exceptional Richardson orbits and confirms a related conjecture, advancing understanding in the representation theory of algebraic groups.
Contribution
It provides a verification of Achar and Sommers' conjecture for these specific orbits, offering new insights into their structure.
Findings
01
Verification of Achar and Sommers' conjecture for exceptional Richardson orbits
02
Identification of special unipotent representations associated with these orbits
03
Enhanced understanding of the representation theory of complex algebraic groups
Abstract
We study special unipotent representations attached to complex exceptional Richardson orbits. As a consequence, we verify a conjecture of Achar and Sommers for these orbits.
Tables5
Equations111
Φ:{(O,σ)∣O⊂N,σ∈(Ge)∨for anye∈O}⟶Λ+(G),
Φ:{(O,σ)∣O⊂N,σ∈(Ge)∨for anye∈O}⟶Λ+(G),
Ψ:{(O,π)∣O⊂Nspecial orbit,π∈(A(O))∨}⟶Λ+(G),
Ψ:{(O,π)∣O⊂Nspecial orbit,π∈(A(O))∨}⟶Λ+(G),
hC≅h×h,tC={(x,−x):x∈h}≅h,aC≅{(x,x):x∈h}≅h.
hC≅h×h,tC={(x,−x):x∈h}≅h,aC≅{(x,x):x∈h}≅h.
{η}:=w′η
{η}:=w′η
X(λL,λR):=IndBG(Cη⊗Cν⊗triv)K−finite,
X(λL,λR):=IndBG(Cη⊗Cν⊗triv)K−finite,
Π(LO):={J(λO,wλO)∣AV(J(λO,wλO))⊆O},
Π(LO):={J(λO,wλO)∣AV(J(λO,wλO))⊆O},
Π(LO)={XO,π∣π∈A(O)∨},
Π(LO)={XO,π∣π∈A(O)∨},
μ:T∗(G/P)≅G×Pn
μ:T∗(G/P)≅G×Pn
A4+A1,D5(a1)inE7,E6(a1)+A1,E7(a3)inE8.
A4+A1,D5(a1)inE7,E6(a1)+A1,E7(a3)inE8.
D4(a1)+A1inE7;D6(a1),D7(a2)inE8
D4(a1)+A1inE7;D6(a1),D7(a2)inE8
ψ∈A(e)∑dimVψ⋅[σO,ψ:IndW(L)W(sgn)],
ψ∈A(e)∑dimVψ⋅[σO,ψ:IndW(L)W(sgn)],
P(O):={wλO∣wλO∣lis dominant and regular}
P(O):={wλO∣wλO∣lis dominant and regular}
XO,πi:=J(λ1,λi)≅IndLG(Vi)
XO,πi:=J(λ1,λi)≅IndLG(Vi)
w∈W(L)∑sgn(w)X(λ1,wλi),
w∈W(L)∑sgn(w)X(λ1,wλi),
XO,π∣KC≅R(O,ψ)≅IndGeG(Wψ),
XO,π∣KC≅R(O,ψ)≅IndGeG(Wψ),
R(O,σ)≅IndGeG(Wσ)≅λ∈Λ+(G)∑mO,σ(λ)IndTG(eλ),
R(O,σ)≅IndGeG(Wσ)≅λ∈Λ+(G)∑mO,σ(λ)IndTG(eλ),
Φ:{(O,σ)∣O⊂N,σ∈(Ge)∨for anye∈O}→Λ+(G)
Φ:{(O,σ)∣O⊂N,σ∈(Ge)∨for anye∈O}→Λ+(G)
SLO:={LP⊆LO∣LP⊈LOspfor other special LOsp⊊LO}.
SLO:={LP⊆LO∣LP⊈LOspfor other special LOsp⊊LO}.
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TopicsAdvanced Algebra and Geometry · Finite Group Theory Research · Advanced Topics in Algebra
Full text
Unipotent Representations of Exceptional Richardson Orbits
Kayue Daniel Wong
School of Science and Engineering, the Chinese University of Hong Kong, Shenzhen, Guangdong 518172, P.R. China
We study special unipotent representations attached to complex exceptional Richardson orbits.
As a consequence, we verify a conjecture of Achar and Sommers for these orbits.
1. Introduction
In [BV3], Barbasch and Vogan studied
special unipotent representations for complex
simple Lie groups G. These representations are
of interest in various areas of representation
theory. For instance, they are conjectured to be the
‘building blocks’ of the unitary dual of G. This idea is verified
by Vogan [V1] for G=GL(n,F) where F=C,R
and H, and by Barbasch [B1] for all complex classical groups.
Moreover, these representations are conjectured
to be quantization models of special nilpotent orbits (Conjecture 2.5).
The conjecture was shown to be true by Barbasch [B4] for classical nilpotent orbits,
and by Losev, Mason-Brown and Matvieievskyi [LMM], [MM] in general
(Remark 2.7).
A main goal of this manuscript is to study a map defined by Lusztig and Vogan
on nilpotent orbits of complex reductive Lie algebras. More explicitly, let N
be the set of nilpotent elements of the Lie algebra g of G,
and O⊂N be a nilpotent orbit. For any
e∈O, let Ge be the stabilizer subgroup of e under conjugation, and
(Ge)∨ be the isomorphism class of irreducible, algebraic representations
of Ge. They conjectured that there is a bijection between the sets
[TABLE]
where Λ+(G)⊂t∗ be the collection of highest dominant
weights of finite dimensional representations of G. The conjecture is later proved by Bezrukavnikov,
and the image of Φ is computed explicitly in some cases – namely,
Achar [Ac] computed the bijection G=GL(n,C), and the calculations are later
simplified by Rush [R]. On the other hand,
Liang-Zhang [LZ] computed the bijection for minimal nilpotent orbits, and
Zhang [Z] computed the case of G=G2(C).
In [W2], the author defined a map analogous to (1):
[TABLE]
where O is a special orbit in the sense of Lusztig, and A(O) is
Lusztig’s quotient of the component group A(O):=Ge/(Ge)0 of Ge (c.f. [L1, Chapters 4, 13]).
To relate Ψ to special unipotent representations and quantization, note that the domain of Ψ
is in one-to-one correspondence with the special unipotent representations of G.
Indeed, given the validity of the quantization conjecture mentioned above,
then Ψ is equal to Φ for π∈(Ge)∨
that factors through Ge→Ge/(Ge)0=A(O)→A(O).
In such cases, one can effectively compute (part of) Φ by understanding Ψ. This idea is pursued in [W2]
for classical nilpotent orbits, so that
one can prove a conjecture by Achar and Sommers [AS] on the image of Φ for classical Lie groups.
In this manuscript, we compute the image of Ψ
explicitly for all complex exceptional Richardson orbits.
As a consequence, we prove that the Achar-Sommers conjecture also holds
for these orbits (Theorem 2.10).
2. Preliminaries
2.1. Special Unipotent Representations
Let G be a complex simple Lie group, with maximal compact subgroup K. We recall the construction
of irreducible, admissible (gC,KC)-modules, where KC≅G
is the complexification of K.
Let H=TA be the Cartan decomposition of the Cartan subgroup H of G, with h=t+a. We make the following identifications:
[TABLE]
Let (λL,λR)∈h∗×h∗ such that η:=λL−λR is integral, and write
[TABLE]
as the unique dominant weight to which η is conjugate under the action of the Weyl group w′∈W:=W(g,h).
Write ν:=λL+λR. We can view η as a weight of T and ν a character of A. Put
[TABLE]
where B⊃H is the Borel subgroup of G determined by a choice of positive roots ΔG+.
Then we define J(λL,λR) be the unique irreducible subquotient of X(λL,λR) containing the KC≅G-type V{η}G. By [Zh], every irreducible admissible (g,K)-module has the form J(λL,λR). We will refer to the pair (λL,λR) as the Zhelobenko parameter for the module J(λL,λR).
Among all J(λL,λR)’s, we focus on the following collection of irreducible modules:
Definition 2.1**.**
Let G be a complex Lie group, and LO⊂Lg be a special nilpotent orbit in the Langlands dual
of g. Writing O as the Lusztig-Spaltenstein dual of LO, and λO:=2LH
as one half of the semisimple element in a Jacobson-Morozov triple attached to LO. Then
the special unipotent representations attached to LO are given by the set
[TABLE]
where AV(X) is the associated variety of any (gC,KC)-module X (c.f. [V3, Section 2]).
By Corollary 5.18 of
[BV3], J(λO,wλO) have associated
variety greater than or equal to O for all w∈W.
Therefore Π(LO) consists of irreducible representations
J(λO,wλO)’s whose associated
varieties are precisely equal to O. Note that by Theorem 1.5
of [BV3], J(λ,wμ)≅J(xλ,xwμ) for all x∈W.
So we can begin with any W-conjugate of λ.
The cardinality of Π(LO) is equal to ∣A(O)∨∣, i.e.
the number of irreducible, non-isomorphic representations of A(O).
Moreover, by writing
[TABLE]
*then the character formula for each XO,π can be obtained explicitly.
*
A detailed proof of Theorem 2.2 is given by [BV3] in the case when λO
is integral, i.e. LO is an even orbit. In the Appendix, we complete the proof of the theorem
for non-integral λO.
2.2. Richardson Orbits
Although one can get the character formula for all XO,π by Theorem 2.2,
its expression can be quite complicated. Therefore, we focus on the case when O is Richardson, i.e,
there exists a parabolic subalgebra p=l+n such that O is the dense
orbit in the G-saturation G⋅n of n.
In particular, their closures can be obtained by the image of the moment map
[TABLE]
given by μ(g,X):=g⋅X for some parabolic group P=LN. We would like to investigate when the above map is birational:
Proposition 2.3**.**
Let O be an exceptional Richardson orbit other than
[TABLE]
Then there exists a parabolic subgroup P such that the moment map μ in (4) is birational onto O.
Proof.
Our choices of P for all Richardson orbits other than the four orbits in (5)
are listed in Sections 4 – 8. It is known that if [Ge:Pe]=1, then μ is birational.
This holds when Ge/(Ge)0=A(O)=1. Also, by the results
in [McG], the map is birational when O
is even (so that the Levi subgroup L is given by the
nodes marked with [math]’s). By checking the tables in
[CM], we are only left with
[TABLE]
along with the four orbits mentioned in the proposition.
Indeed, we can apply Lemma 5.5 in [FJLS] to check birationality
of μ, which says that for any e∈O,
the number of components in μ−1(e) is equal to
[TABLE]
where σO,ψ is the Springer
representation Htop(Be)ψ.
One can use the table in [Alv] to check that the above value
is equal to 1 for the orbits in (6).
Therefore μ is birational for all the orbits stated in the proposition.
∎
For all exceptional Richardson O other than the four orbits in (5),
Theorem 9.11 of [BV3] provides a nice character formula for all Π(LO).
Theorem 2.4**.**
Let O be an exceptional Richardson orbit not equal to four orbits in (5), and l be
the Levi subalgebra of the parabolic subgroup corresponding to O given by Proposition 2.3, then the following statements hold:
(a)
The number of elements in
[TABLE]
is equal to the number of conjugacy classes of A(O). In particular, there exists a unique λ1 such that (λ1,α∨)=1 for all simple roots in l, and λ1−λ is a sum of positive roots for all λ∈P(O).
(b)
The elements in P(O)={λ1,λ2,…,λr} can be arranged such that ∣λi−λ1∣<∣λi+1−λ1∣ for all i.
(c)
The representations J(λ1,λi) exhaust all special unipotent representations in Π(LO). More precisely, suppose A(O)≅Sk for 2≤k≤5, and C1=⟨k⟩>⋯>Cr=⟨1k⟩ is the total ordering of partitions of k (such ordering exists since k≤5). Then πi∈A(O)∨
are parametrized by the partition Ci, with
[TABLE]
for some finite-dimensional representations Vi of L with highest weight (λ1−λi)∣l.
In particular, when i=1, then π1 is the trivial representation of A(O)∧ and V1 is the trivial representation of L.
Proof.
The values of λi in (a) and (b) are computed in Sections 4 – 8. Assuming the results of (a) and (b) hold, then for each λi, let Ii be as defined in Section 9 of [BV3] such that the character formula of
Ii is given by
[TABLE]
so that Ii≅IndLG(Vλ1−λiL) as KC≅G-modules. Obviously, we have AV(Ii)=Indlg(0)=O, and its composition factors must consist of special unipotent representations attached to O. Using the arguments in 9.11 - 9.21 of [BV3], one can see Ii=J(λ1,λi) and (c) follows.
∎
2.3. Vogan’s Conjecture on Quantization
As mentioned in the introduction, the special unipotent representations Π(LO)
are conjectured to be the ‘quantizations’ of (local systems of) O. More precisely, we
have the following conjecture by Vogan in the 1980s:
Let O be a complex special nilpotent orbit, and
π be an irreducible representation of A(O).
Then there exists an irreducible representation Wψ
of Ge that factors through Ge/(Ge)0=A(O) such that:
[TABLE]
where R(O,ψ) is the global section of
the vector bundle G×GeWψ→G/Ge≅O.
One can use the results in [S3] to see that
the irreducible L-modules Vi in Theorem 2.4(c) are all ‘lifts’
of some Wψi∈A(O)∧ in Conjecture
2.5. Namely, the restricted representation Vi∣Le factors through
Le/(Le)0≅Ge/(Ge)0, and is isomorphic to Wψi.
This gives an evidence on the validity of Conjecture 2.5.
Indeed, for triv∈A(O)∨, one can easily verify the conjecture for
Richardson orbits:
Theorem 2.6**.**
Let O be an exceptional Richardson orbit. Then XO,triv≅R(O), i.e. Conjecture 2.5
holds for all exceptional Richardson orbits.
Proof.
When O is not equal to the four orbits in (5), the map μ:T∗(G/P)→O is the normalization of the orbit closure O. By standard arguments in algebraic geometry (see [J] for example), R(O)≅C[T∗(G/P)]. By a result of [McG], the latter is isomorphic to IndLG(triv) as G-modules. This means R(O)≅IndLG(triv)≅XO,triv and therefore Theorem 2.6 holds for these orbits. We will get the same conclusion for the four orbits in (5) in Section 9.
∎
Remark 2.7**.**
As mentioned in the introduction, Conjecture 2.5 is proved in full generality
in [LMM] and [MM]. More explicitly, a definition of unipotent representations
for G-equivariant covers of nilpotent orbits is given in [LMM, Definition 1.4.1].
It is shown in [LMM] (for classical groups) and [MM] (for exceptional and spin groups)
that all special unipotent representations XO,π in Definition 2.1
are unipotent representations attached to some covers
of O. Then by [LMM, Section 6.7], Conjecture 2.5 holds for all
special unipotent representations.
2.4. The Lusztig-Vogan map
We now give a precise description of the map (1) given by Theorem 8.2 of [V4]:
Definition 2.8**.**
For each nilpotent element O⊂g, and Wσ∈(Ge)∨ for any e∈O, write
[TABLE]
where all but finitely many mO,σ(λ)∈Z are zero. Then the Lusztig-Vogan map
[TABLE]
is defined by Φ(O,σ)=λmax,
where λmax is the largest dominant element in
Equation (8) such that mO,σ(λmax)=0.
The paper [W2] studies Φ
for all classical nilpotent orbits O, and all (Ge)∨ that factors through (Ge/(Ge)0)∨=A(O)∨,
and consequently proved the
following conjecture of Achar and Sommers for classical groups:
Let O be a special nilpotent orbit. Consider the special pieceSLO ([L2]) of the Lusztig-Spaltenstein dual LO of O:
[TABLE]
For each LP∈SLO, let HLP be the semi-simple element of a Jacobson-Morozov triple
corresponding to LP. Then there exists Wψ∈A(O)∧ (which can be seen as a representation of
Ge that factors through A(O)) such that
[TABLE]
Let us describe how the above conjecture is proved in [W2] for classical groups. In [W2, Definition 1.5], the author defined the map
[TABLE]
as follows: Suppose the character formula XO,π is given by
[TABLE]
On the level of Grothendieck groups, the restriction of XO,π into KC-modules is of the form
[TABLE]
where {λO−wλO} is as defined in (2), and bO,π(λ)∈Z. Then we define Ψ(O,π):=λmax,
where λmax is the largest dominant element in
(10) such that bO,π(λmax)= [math].
Since Conjecture 2.5 holds for all nilpotent orbits,
one can relate the two maps Ψ and Φ by Equations (8)
and (10). As a consequence, one can verify Conjecture 2.9
by computing Ψ. This is done for all classical special nilpotent orbits O
and all π∈A(O)∨ in [W2]. In this manuscript, we apply the same strategy for exceptional Richardson orbits and obtain the following:
Theorem 2.10**.**
Let G be a complex exceptional Lie group of adjoint type,
and O be a Richardson orbit. For every semisimple
element HLP in a Jacobson-Morozov triple
corresponding to LP∈SLO, there exists π∈A(O)∧
such that
[TABLE]
Consequently, Conjecture 2.9 holds for all exceptional Richardson orbits.
In Sections 4 – 8, we write down the details for Theorem 2.4 and Theorem 2.10 for all Richardson orbits other than the four orbits in
(5). Namely, for each exceptional Richardson orbit O, we list
•
the Levi subalgebra l where O is induced from, by specifying a sub-diagram of the Dynkin diagram of g;
•
all irreducible representations πi of A(O)≅Sk (k=2,3,4,5) in terms of partitions of k;
•
the values of λi appearing in Theorem 2.4(a)–(b).
We now turn our attention to Theorem 2.10. By Equation (7) and (10), the image of Ψ can be easily computed as
[TABLE]
for all πi∈A(O)∨, where wL is the longest element in the Weyl group W(L). The values of Ψ(O,πi) are recorded on the second last column of the tables in terms of the weighted Dynkin diagram of Lg. And the last column records the orbit LP whose Dynkin element HLP is given by the previous column, which verifies Theorem 2.10 for all but four exceptional Richardson orbits given in (5).
4. G2 Orbits
The exceptional group G2 has three Richardson orbits. Fix the Dynkin diagram of g by: (-2,1,1)$$(1,-1,0)$$3
.
5. F4 Orbits
The exceptional group F4 has nine Richardson orbits. Fix the Dynkin diagram of g by
12234
with simple roots:
[TABLE]
6. E6 Orbits
The adjoint exceptional group E6 has fifteen Richardson orbits. Fix the Dynkin diagram
654231
with simple roots:
[TABLE]
7. E7 Orbits
The adjoint exceptional group E7 has twenty seven Richardson orbits excluding A4+A1 and D5(a1). Fix the Dynkin diagram
7654231
with simple roots:
[TABLE]
8. E8 Orbits
The exceptional group E8 has thirty two Richardson orbits excluding E6(a1)+A1 and E7(a3). Fix the Dynkin diagram
87654231
with simple roots:
[TABLE]
9. Proof of Theorem 2.10 - the four exceptional cases
We now finish the proof of Theorem 2.10 for
O equal to the four orbits in (5). Note that all these
orbits have Lusztig’s quotient A(O)≅S2, and the special piece
SLO={LO} only contains one element. Therefore, one only
needs to find one π∈A(O)∨ such that Ψ(O,π)=HLO
in order to verify Theorem 2.10.
Note that the orbits A4+A1 in E7 and E6(a1)+A1
in E8 are called exceptional in Section 4 of [BV3].
9.1. O=A4+A1 in E7
By checking the tables of [AL] directly
(the calculations for other orbits can be found in the Appendix), the two left cell representations attached to O are equal to JD6+A1E7(σ1), JD6+A1E7(σ2), where σ1, σ2 are the two left cell representations attached to O′=[332211]+[11] in D6+A1. Using Proposition 6.6 of [BV3], the character formulas for XO,π can be derived from that of O′ in
[TABLE]
Using Theorem 3.4 in [W2], Ψ(O′,⟨2⟩)=\leavevmodeto61.46pt\vboxto33.73pt\pgfpicture\makeatletter\lower-13.31044ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfontto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\definecolor[named]pgffillcolorrgb0,0,0\pgfsys@color@gray@fill0\pgfsys@invoke\pgfsys@moveto0.0pt0.0pt\pgfsys@moveto1.42271pt0.0pt\pgfsys@curveto1.42271pt0.78575pt0.78575pt1.42271pt0.0pt1.42271pt\pgfsys@curveto-0.78575pt1.42271pt-1.42271pt0.78575pt-1.42271pt0.0pt\pgfsys@curveto-1.42271pt-0.78575pt-0.78575pt-1.42271pt0.0pt-1.42271pt\pgfsys@curveto0.78575pt-1.42271pt1.42271pt-0.78575pt1.42271pt0.0pt\pgfsys@closepath\pgfsys@moveto0.0pt0.0pt\pgfsys@lineto8.5359pt0.0pt\pgfsys@moveto9.95862pt0.0pt\pgfsys@curveto9.95862pt0.78575pt9.32166pt1.42271pt8.5359pt1.42271pt\pgfsys@curveto7.75015pt1.42271pt7.11319pt0.78575pt7.11319pt0.0pt\pgfsys@curveto7.11319pt-0.78575pt7.75015pt-1.42271pt8.5359pt-1.42271pt\pgfsys@curveto9.32166pt-1.42271pt9.95862pt-0.78575pt9.95862pt0.0pt\pgfsys@closepath\pgfsys@moveto8.5359pt0.0pt\pgfsys@lineto17.07182pt0.0pt\pgfsys@moveto18.49454pt0.0pt\pgfsys@curveto18.49454pt0.78575pt17.85757pt1.42271pt17.07182pt1.42271pt\pgfsys@curveto16.28607pt1.42271pt15.64911pt0.78575pt15.64911pt0.0pt\pgfsys@curveto15.64911pt-0.78575pt16.28607pt-1.42271pt17.07182pt-1.42271pt\pgfsys@curveto17.85757pt-1.42271pt18.49454pt-0.78575pt18.49454pt0.0pt\pgfsys@closepath\pgfsys@moveto17.07182pt0.0pt\pgfsys@lineto25.6073pt0.0pt\pgfsys@moveto27.03001pt0.0pt\pgfsys@curveto27.03001pt0.78575pt26.39305pt1.42271pt25.6073pt1.42271pt\pgfsys@curveto24.82155pt1.42271pt24.18459pt0.78575pt24.18459pt0.0pt\pgfsys@curveto24.18459pt-0.78575pt24.82155pt-1.42271pt25.6073pt-1.42271pt\pgfsys@curveto26.39305pt-1.42271pt27.03001pt-0.78575pt27.03001pt0.0pt\pgfsys@closepath\pgfsys@moveto25.6073pt0.0pt\pgfsys@lineto25.6073pt0.0pt\pgfsys@lineto25.6073pt7.11319pt\pgfsys@moveto27.03001pt7.11319pt\pgfsys@curveto27.03001pt7.89894pt26.39305pt8.5359pt25.6073pt8.5359pt\pgfsys@curveto24.82155pt8.5359pt24.18459pt7.89894pt24.18459pt7.11319pt\pgfsys@curveto24.18459pt6.32744pt24.82155pt5.69048pt25.6073pt5.69048pt\pgfsys@curveto26.39305pt5.69048pt27.03001pt6.32744pt27.03001pt7.11319pt\pgfsys@closepath\pgfsys@moveto25.6073pt7.11319pt\pgfsys@moveto25.6073pt0.0pt\pgfsys@lineto34.1432pt0.0pt\pgfsys@moveto35.56592pt0.0pt\pgfsys@curveto35.56592pt0.78575pt34.92896pt1.42271pt34.1432pt1.42271pt\pgfsys@curveto33.35745pt1.42271pt32.72049pt0.78575pt32.72049pt0.0pt\pgfsys@curveto32.72049pt-0.78575pt33.35745pt-1.42271pt34.1432pt-1.42271pt\pgfsys@curveto34.92896pt-1.42271pt35.56592pt-0.78575pt35.56592pt0.0pt\pgfsys@closepath\pgfsys@moveto34.1432pt0.0pt\pgfsys@moveto49.79233pt0.0pt\pgfsys@moveto51.21504pt0.0pt\pgfsys@curveto51.21504pt0.78575pt50.57808pt1.42271pt49.79233pt1.42271pt\pgfsys@curveto49.00658pt1.42271pt48.36961pt0.78575pt48.36961pt0.0pt\pgfsys@curveto48.36961pt-0.78575pt49.00658pt-1.42271pt49.79233pt-1.42271pt\pgfsys@curveto50.57808pt-1.42271pt51.21504pt-0.78575pt51.21504pt0.0pt\pgfsys@closepath\pgfsys@moveto49.79233pt0.0pt\pgfsys@fillstroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.0-2.5pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke0\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.06.0359pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.014.57182pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke0\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.023.1073pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.023.1073pt10.6462pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke0\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.031.6432pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke0\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.047.29233pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture;Ψ(O′,⟨12⟩)=\leavevmodeto61.46pt\vboxto33.73pt\pgfpicture\makeatletter\lower-13.31044ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfontto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\definecolor[named]pgffillcolorrgb0,0,0\pgfsys@color@gray@fill0\pgfsys@invoke\pgfsys@moveto0.0pt0.0pt\pgfsys@moveto1.42271pt0.0pt\pgfsys@curveto1.42271pt0.78575pt0.78575pt1.42271pt0.0pt1.42271pt\pgfsys@curveto-0.78575pt1.42271pt-1.42271pt0.78575pt-1.42271pt0.0pt\pgfsys@curveto-1.42271pt-0.78575pt-0.78575pt-1.42271pt0.0pt-1.42271pt\pgfsys@curveto0.78575pt-1.42271pt1.42271pt-0.78575pt1.42271pt0.0pt\pgfsys@closepath\pgfsys@moveto0.0pt0.0pt\pgfsys@lineto8.5359pt0.0pt\pgfsys@moveto9.95862pt0.0pt\pgfsys@curveto9.95862pt0.78575pt9.32166pt1.42271pt8.5359pt1.42271pt\pgfsys@curveto7.75015pt1.42271pt7.11319pt0.78575pt7.11319pt0.0pt\pgfsys@curveto7.11319pt-0.78575pt7.75015pt-1.42271pt8.5359pt-1.42271pt\pgfsys@curveto9.32166pt-1.42271pt9.95862pt-0.78575pt9.95862pt0.0pt\pgfsys@closepath\pgfsys@moveto8.5359pt0.0pt\pgfsys@lineto17.07182pt0.0pt\pgfsys@moveto18.49454pt0.0pt\pgfsys@curveto18.49454pt0.78575pt17.85757pt1.42271pt17.07182pt1.42271pt\pgfsys@curveto16.28607pt1.42271pt15.64911pt0.78575pt15.64911pt0.0pt\pgfsys@curveto15.64911pt-0.78575pt16.28607pt-1.42271pt17.07182pt-1.42271pt\pgfsys@curveto17.85757pt-1.42271pt18.49454pt-0.78575pt18.49454pt0.0pt\pgfsys@closepath\pgfsys@moveto17.07182pt0.0pt\pgfsys@lineto25.6073pt0.0pt\pgfsys@moveto27.03001pt0.0pt\pgfsys@curveto27.03001pt0.78575pt26.39305pt1.42271pt25.6073pt1.42271pt\pgfsys@curveto24.82155pt1.42271pt24.18459pt0.78575pt24.18459pt0.0pt\pgfsys@curveto24.18459pt-0.78575pt24.82155pt-1.42271pt25.6073pt-1.42271pt\pgfsys@curveto26.39305pt-1.42271pt27.03001pt-0.78575pt27.03001pt0.0pt\pgfsys@closepath\pgfsys@moveto25.6073pt0.0pt\pgfsys@lineto25.6073pt0.0pt\pgfsys@lineto25.6073pt7.11319pt\pgfsys@moveto27.03001pt7.11319pt\pgfsys@curveto27.03001pt7.89894pt26.39305pt8.5359pt25.6073pt8.5359pt\pgfsys@curveto24.82155pt8.5359pt24.18459pt7.89894pt24.18459pt7.11319pt\pgfsys@curveto24.18459pt6.32744pt24.82155pt5.69048pt25.6073pt5.69048pt\pgfsys@curveto26.39305pt5.69048pt27.03001pt6.32744pt27.03001pt7.11319pt\pgfsys@closepath\pgfsys@moveto25.6073pt7.11319pt\pgfsys@moveto25.6073pt0.0pt\pgfsys@lineto34.1432pt0.0pt\pgfsys@moveto35.56592pt0.0pt\pgfsys@curveto35.56592pt0.78575pt34.92896pt1.42271pt34.1432pt1.42271pt\pgfsys@curveto33.35745pt1.42271pt32.72049pt0.78575pt32.72049pt0.0pt\pgfsys@curveto32.72049pt-0.78575pt33.35745pt-1.42271pt34.1432pt-1.42271pt\pgfsys@curveto34.92896pt-1.42271pt35.56592pt-0.78575pt35.56592pt0.0pt\pgfsys@closepath\pgfsys@moveto34.1432pt0.0pt\pgfsys@moveto49.79233pt0.0pt\pgfsys@moveto51.21504pt0.0pt\pgfsys@curveto51.21504pt0.78575pt50.57808pt1.42271pt49.79233pt1.42271pt\pgfsys@curveto49.00658pt1.42271pt48.36961pt0.78575pt48.36961pt0.0pt\pgfsys@curveto48.36961pt-0.78575pt49.00658pt-1.42271pt49.79233pt-1.42271pt\pgfsys@curveto50.57808pt-1.42271pt51.21504pt-0.78575pt51.21504pt0.0pt\pgfsys@closepath\pgfsys@moveto49.79233pt0.0pt\pgfsys@fillstroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.0-2.5pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke0\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.06.0359pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.014.57182pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.023.1073pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke0\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.023.1073pt10.6462pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.031.6432pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.047.29233pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture. Therefore, one can compute that
[TABLE]
[TABLE]
In terms of Dynkin diagram, we have
[TABLE]
Note that we have Ψ(A4+A1,⟨2⟩)=HA4+A1=HL(A4+A1) and Theorem 2.10 holds.
9.2. O=D5(a1) in E7
As in the above subsection, one can derive the character formulas of XO,π from that of O′=[332211] in D6. More precisely, we have
Ψ(O′,⟨2⟩)=\leavevmodeto61.46pt\vboxto33.73pt\pgfpicture\makeatletter\lower-13.31044ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfontto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\definecolor[named]pgffillcolorrgb0,0,0\pgfsys@color@gray@fill0\pgfsys@invoke\pgfsys@moveto0.0pt0.0pt\pgfsys@moveto1.42271pt0.0pt\pgfsys@curveto1.42271pt0.78575pt0.78575pt1.42271pt0.0pt1.42271pt\pgfsys@curveto-0.78575pt1.42271pt-1.42271pt0.78575pt-1.42271pt0.0pt\pgfsys@curveto-1.42271pt-0.78575pt-0.78575pt-1.42271pt0.0pt-1.42271pt\pgfsys@curveto0.78575pt-1.42271pt1.42271pt-0.78575pt1.42271pt0.0pt\pgfsys@closepath\pgfsys@moveto0.0pt0.0pt\pgfsys@lineto8.5359pt0.0pt\pgfsys@moveto9.95862pt0.0pt\pgfsys@curveto9.95862pt0.78575pt9.32166pt1.42271pt8.5359pt1.42271pt\pgfsys@curveto7.75015pt1.42271pt7.11319pt0.78575pt7.11319pt0.0pt\pgfsys@curveto7.11319pt-0.78575pt7.75015pt-1.42271pt8.5359pt-1.42271pt\pgfsys@curveto9.32166pt-1.42271pt9.95862pt-0.78575pt9.95862pt0.0pt\pgfsys@closepath\pgfsys@moveto8.5359pt0.0pt\pgfsys@lineto17.07182pt0.0pt\pgfsys@moveto18.49454pt0.0pt\pgfsys@curveto18.49454pt0.78575pt17.85757pt1.42271pt17.07182pt1.42271pt\pgfsys@curveto16.28607pt1.42271pt15.64911pt0.78575pt15.64911pt0.0pt\pgfsys@curveto15.64911pt-0.78575pt16.28607pt-1.42271pt17.07182pt-1.42271pt\pgfsys@curveto17.85757pt-1.42271pt18.49454pt-0.78575pt18.49454pt0.0pt\pgfsys@closepath\pgfsys@moveto17.07182pt0.0pt\pgfsys@lineto25.6073pt0.0pt\pgfsys@moveto27.03001pt0.0pt\pgfsys@curveto27.03001pt0.78575pt26.39305pt1.42271pt25.6073pt1.42271pt\pgfsys@curveto24.82155pt1.42271pt24.18459pt0.78575pt24.18459pt0.0pt\pgfsys@curveto24.18459pt-0.78575pt24.82155pt-1.42271pt25.6073pt-1.42271pt\pgfsys@curveto26.39305pt-1.42271pt27.03001pt-0.78575pt27.03001pt0.0pt\pgfsys@closepath\pgfsys@moveto25.6073pt0.0pt\pgfsys@lineto25.6073pt0.0pt\pgfsys@lineto25.6073pt7.11319pt\pgfsys@moveto27.03001pt7.11319pt\pgfsys@curveto27.03001pt7.89894pt26.39305pt8.5359pt25.6073pt8.5359pt\pgfsys@curveto24.82155pt8.5359pt24.18459pt7.89894pt24.18459pt7.11319pt\pgfsys@curveto24.18459pt6.32744pt24.82155pt5.69048pt25.6073pt5.69048pt\pgfsys@curveto26.39305pt5.69048pt27.03001pt6.32744pt27.03001pt7.11319pt\pgfsys@closepath\pgfsys@moveto25.6073pt7.11319pt\pgfsys@moveto25.6073pt0.0pt\pgfsys@lineto34.1432pt0.0pt\pgfsys@moveto35.56592pt0.0pt\pgfsys@curveto35.56592pt0.78575pt34.92896pt1.42271pt34.1432pt1.42271pt\pgfsys@curveto33.35745pt1.42271pt32.72049pt0.78575pt32.72049pt0.0pt\pgfsys@curveto32.72049pt-0.78575pt33.35745pt-1.42271pt34.1432pt-1.42271pt\pgfsys@curveto34.92896pt-1.42271pt35.56592pt-0.78575pt35.56592pt0.0pt\pgfsys@closepath\pgfsys@moveto34.1432pt0.0pt\pgfsys@moveto49.79233pt0.0pt\pgfsys@moveto51.21504pt0.0pt\pgfsys@curveto51.21504pt0.78575pt50.57808pt1.42271pt49.79233pt1.42271pt\pgfsys@curveto49.00658pt1.42271pt48.36961pt0.78575pt48.36961pt0.0pt\pgfsys@curveto48.36961pt-0.78575pt49.00658pt-1.42271pt49.79233pt-1.42271pt\pgfsys@curveto50.57808pt-1.42271pt51.21504pt-0.78575pt51.21504pt0.0pt\pgfsys@closepath\pgfsys@moveto49.79233pt0.0pt\pgfsys@fillstroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.0-2.5pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke0\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.06.0359pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.014.57182pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke0\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.023.1073pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.023.1073pt10.6462pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke0\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.031.6432pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke0\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.047.29233pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke0\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture;Ψ(O′,⟨12⟩)=\leavevmodeto61.46pt\vboxto33.73pt\pgfpicture\makeatletter\lower-13.31044ptto0.0pt\pgfsys@beginscope\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke\pgfsys@setlinewidth0.4pt\pgfsys@invoke\nullfontto0.0pt\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\definecolor[named]pgffillcolorrgb0,0,0\pgfsys@color@gray@fill0\pgfsys@invoke\pgfsys@moveto0.0pt0.0pt\pgfsys@moveto1.42271pt0.0pt\pgfsys@curveto1.42271pt0.78575pt0.78575pt1.42271pt0.0pt1.42271pt\pgfsys@curveto-0.78575pt1.42271pt-1.42271pt0.78575pt-1.42271pt0.0pt\pgfsys@curveto-1.42271pt-0.78575pt-0.78575pt-1.42271pt0.0pt-1.42271pt\pgfsys@curveto0.78575pt-1.42271pt1.42271pt-0.78575pt1.42271pt0.0pt\pgfsys@closepath\pgfsys@moveto0.0pt0.0pt\pgfsys@lineto8.5359pt0.0pt\pgfsys@moveto9.95862pt0.0pt\pgfsys@curveto9.95862pt0.78575pt9.32166pt1.42271pt8.5359pt1.42271pt\pgfsys@curveto7.75015pt1.42271pt7.11319pt0.78575pt7.11319pt0.0pt\pgfsys@curveto7.11319pt-0.78575pt7.75015pt-1.42271pt8.5359pt-1.42271pt\pgfsys@curveto9.32166pt-1.42271pt9.95862pt-0.78575pt9.95862pt0.0pt\pgfsys@closepath\pgfsys@moveto8.5359pt0.0pt\pgfsys@lineto17.07182pt0.0pt\pgfsys@moveto18.49454pt0.0pt\pgfsys@curveto18.49454pt0.78575pt17.85757pt1.42271pt17.07182pt1.42271pt\pgfsys@curveto16.28607pt1.42271pt15.64911pt0.78575pt15.64911pt0.0pt\pgfsys@curveto15.64911pt-0.78575pt16.28607pt-1.42271pt17.07182pt-1.42271pt\pgfsys@curveto17.85757pt-1.42271pt18.49454pt-0.78575pt18.49454pt0.0pt\pgfsys@closepath\pgfsys@moveto17.07182pt0.0pt\pgfsys@lineto25.6073pt0.0pt\pgfsys@moveto27.03001pt0.0pt\pgfsys@curveto27.03001pt0.78575pt26.39305pt1.42271pt25.6073pt1.42271pt\pgfsys@curveto24.82155pt1.42271pt24.18459pt0.78575pt24.18459pt0.0pt\pgfsys@curveto24.18459pt-0.78575pt24.82155pt-1.42271pt25.6073pt-1.42271pt\pgfsys@curveto26.39305pt-1.42271pt27.03001pt-0.78575pt27.03001pt0.0pt\pgfsys@closepath\pgfsys@moveto25.6073pt0.0pt\pgfsys@lineto25.6073pt0.0pt\pgfsys@lineto25.6073pt7.11319pt\pgfsys@moveto27.03001pt7.11319pt\pgfsys@curveto27.03001pt7.89894pt26.39305pt8.5359pt25.6073pt8.5359pt\pgfsys@curveto24.82155pt8.5359pt24.18459pt7.89894pt24.18459pt7.11319pt\pgfsys@curveto24.18459pt6.32744pt24.82155pt5.69048pt25.6073pt5.69048pt\pgfsys@curveto26.39305pt5.69048pt27.03001pt6.32744pt27.03001pt7.11319pt\pgfsys@closepath\pgfsys@moveto25.6073pt7.11319pt\pgfsys@moveto25.6073pt0.0pt\pgfsys@lineto34.1432pt0.0pt\pgfsys@moveto35.56592pt0.0pt\pgfsys@curveto35.56592pt0.78575pt34.92896pt1.42271pt34.1432pt1.42271pt\pgfsys@curveto33.35745pt1.42271pt32.72049pt0.78575pt32.72049pt0.0pt\pgfsys@curveto32.72049pt-0.78575pt33.35745pt-1.42271pt34.1432pt-1.42271pt\pgfsys@curveto34.92896pt-1.42271pt35.56592pt-0.78575pt35.56592pt0.0pt\pgfsys@closepath\pgfsys@moveto34.1432pt0.0pt\pgfsys@moveto49.79233pt0.0pt\pgfsys@moveto51.21504pt0.0pt\pgfsys@curveto51.21504pt0.78575pt50.57808pt1.42271pt49.79233pt1.42271pt\pgfsys@curveto49.00658pt1.42271pt48.36961pt0.78575pt48.36961pt0.0pt\pgfsys@curveto48.36961pt-0.78575pt49.00658pt-1.42271pt49.79233pt-1.42271pt\pgfsys@curveto50.57808pt-1.42271pt51.21504pt-0.78575pt51.21504pt0.0pt\pgfsys@closepath\pgfsys@moveto49.79233pt0.0pt\pgfsys@fillstroke\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.0-2.5pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke0\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.06.0359pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke2\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.014.57182pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.023.1073pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke0\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.023.1073pt10.6462pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.031.6432pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke1\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke\pgfsys@beginscope\pgfsys@invoke\pgfsys@transformcm1.00.00.01.047.29233pt-9.97743pt\pgfsys@invoke\definecolorpgfstrokecolorrgb0,0,0\pgfsys@color@rgb@stroke000\pgfsys@invoke\pgfsys@color@rgb@fill000\pgfsys@invoke0\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\pgfsys@discardpath\pgfsys@invoke\lxSVG@closescope\pgfsys@endscope\hss\lxSVG@closescope\endpgfpicture. Therefore, one can compute that
[TABLE]
[TABLE]
In terms of Dynkin diagram, we have
[TABLE]
Note that Ψ(D5(a1),⟨2⟩)=HA4=HL(D5(a1)), and Theorem 2.10 holds.
9.3. O=E6(a1)+A1 in E8
As before, the character formulas for XO,π can be derived from that of O′=D5(a1)+[11] in E7+A1, which, by last subsection, can be derived from that of [332211]+[11] in D6+A1. Using the same argument as above, we have
[TABLE]
We have Ψ(E6(a1)+A1,⟨2⟩)=HA4+A1=HL(E6(a1)+A1) and Theorem 2.10 follows.
9.4. O=E7(a3) in E8
Finally, we can derive the character formula of XO,π from that of O′=[332211] in D6. As in the previous subsections, we have
[TABLE]
We have Ψ(E7(a3),⟨2⟩)=HA4=HL(E7(A3)), and Theorem 2.10 holds.
*For the four orbits above, the sum of the two special
unipotent representations attached to A4+A1, D5(a1)
in G=E7, and E6(a1)+A1, E7(a3) in G=E8
are isomorphic to IndLG(triv), where the
semisimple part of L is of type A4+A1 for the orbits
A4+A1, E6(a1)+A1, and of type A4 for D5(a1),
E7(a3). In fact, the orbit [332211] in D6 that shows
up in all the above examples is also a Richardson orbit, induced
from GL(5)×SO(2). The sum of the two special
unipotent representations attached to [332211] is
isomorphic to IndGL(5)×SO(2)SO(12)(triv).
There are two consequences of this observation:
(a)
Non-birationality of μ: We give an
alternative proof of Proposition 2.3
for these four orbits – For example, suppose on the
contrary that μ:T∗(G/P)→D5(a1)
is birational with the semisimple part of L is of
Type A4, then R(D5(a1))≅C[T∗(G/P)]≅IndLE7(triv)≅XD5(a1),triv⊕XD5(a1),sgn.
However, the multiplicity of D5(a1)
in R(D5(a1)) is one, while the multiplicity
in XD5(a1),triv⊕XD5(a1),sgn is greater than one.
(b)
An alternative proof of Conjecture 2.5 for these orbits: By **[V3]**,
we have XO,triv=R(O)−Y1 and XO,sgn=R(O,sgn)−Y2 for some genuine KC-modules Y1, Y2.
Therefore,
[TABLE]
By Proposition 4.6.1 of **[B4]**, the left hand side is equal to R(O)+R(O,sgn). So Y1=Y2=0 and hence we have XO,triv≅R(O) and XO,sgn≅R(O,sgn).
The method we used in (b) in the above remark
can also be applied to verify Conjecture 2.5
for other Richardson orbits with non-birational
moment maps. For example, let O=F4(a3)
in F4. If we take the the parabolic subgroup
whose Levi is of type A2+A1,
then one can show that XO,⟨31⟩≅R(O,⟨31⟩). On the other hand,
if we take the the parabolic subgroup whose Levi
is of type B2, then one can show that XO,⟨22⟩≅R(O,⟨22⟩).
Funding: The author is supported by
Shenzhen Science and Technology Innovation Committee grant
(no. 20220818094918001).
Appendix A Special Unipotent Representations for Non-even LO
Based on the results of [BV3] on the structure of Π(LO)
for even LO, we give a sketch proof of Theorem 2.2 when the special orbit LO is not even
(or equivalently λO is not integral).
Although it is known by the experts that the results in [BV3] carry over to all λO
regardless of its integrality, we do not find a proof of such result in the literature.
Let λ:=λO. Consider the Lie subalgebra g′
of g with roots α satisfying ⟨α∨,λ⟩∈Z.
It turns out that there is a choice of simple roots of g′
such that their inner products with λ
is equal to either [math] or 1.
(II)
Upon restricting to g′, the result in [BV3] applies,
and we can obtain integral special unipotent representations with
infinitesimal character λ=2LH′
for some even orbit LO′∈Lg′.
(III)
More explicitly, let O′∈g′ be the Lusztig-Spaltenstein
dual of LO′, and VL(O′) be the left cell representation
of W′ denoted by VL(w0wO′) in [BV3]. By
Chapter 4 of [L1], the irreducible representations in
VL(O′) are parametrized by {σx′∣x∈[A(O′)]}
(i.e. partitions [n1m1n2m2…] of k if A(O′)≅Sk).
By Theorem III of [BV3], the unipotent
representations Π(LO′) are parametrized by π∈A(O′)∨, and their character formulas
are given by
[TABLE]
Under the above identification of elements in VL(O′), the
representation σsp′:=σ[1k]′ is special, and is the Springer
representation of O′. All other irreducible representations JG′(λ,wλ)
with our choice of λ have character formulas involving special representations τsp′∈(W′)∨,
where deg(τsp′)<deg(σsp′).
(IV)
By a version of the Kazhdan-Lusztig conjecture (c.f. [B1, Theorem 4.2]),
[TABLE]
are character formulas of irreducible representations of G for all π∈A(O′)∨.
The same result goes for character formulas of other irreducible representations JG′(λ,wλ)
that involves τsp′∈(W′)∨.
(V)
Let JW′W(σsp′) is the J-truncated induction defined in [L1, Section 4.3].
It contains a unique special representation σsp
with deg(σsp)=deg(σsp′) (and similarly we have
τsp∈JW′W(τsp′) for all special representations τsp′∈(W′)∨). By studying the W×W-module
C[W] (c.f. [BV3, Proposition 6.6]), these character
formulas in G contain expressions involving σsp (or
τsp), and their associated varieties can be determined
by the Springer correspondence of σsp (or τsp).
Since deg(τsp)<deg(σsp),
the irreducible representations J(λ,wπλ)
in (11) are the ones with smallest possible associated variety
for our choice of λ.
(VI)
We verify the following statements in the next couple of sections:
∙
For all orbits we are studying,
A(O)≅A(O′),
and σsp=JW′W(σsp′) is non-zero and irreducible.
More precisely, σsp is the Springer representation
attached to O, the Lusztig-Spaltenstein
dual of LO.
∙
If O is not equal to
the three exceptional orbits (A4+A1 in E7, A4+A1 and E6(a1)+A1
in E8) listed in [BV3, Definition 4.5], JW′W(σx′) is non-zero and irreducible
for all σx′∈VL(O′). Since VL(O)=JW′W(VL(O′)) by [BV2, Proposition 3.18],
the elements of VL(O) can be parametrized by
[TABLE]
This identification of elements in VL(O)
matches with that of [L1].
The first point guarantees that all J(λ,wπλ)
have associated variety equal to O
(see also Theorem 5.2 of [McG2]), and they have
the same cardinality as the number of
irreducible representations of A(O), i.e. all J(λ,wπλ)∈Π(LO)
and ∣Π(LO)∣=∣A(O)∨∣.
The second point and [BV3, Proposition 6.6]
reformulate Equation (11) into
[TABLE]
In other words, the character formula of J(λ,wπλ)
is given precisely by Theorem III of [BV3], and the result follows.
A.1. Classical Lie algebras
We apply the strategy in the previous section to classify all classical special unipotent representations. Recall from [CM]
that each classical nilpotent orbit LO⊂Lg can be uniquely characterized by
a partition, except in Type D that there are two orbits corresponding to a single
very even partition of the form
[2a≥2a≥2b≥2b≥⋯≥2c≥2c] (the very even orbits). Since all very even orbits are even,
the results in [BV3] can be applied directly.
For convenience, we denote all non-very even LO by their corresponding partition.
Lg** of Type B:*** Let LO=[r2k≥r2k−1≥⋯≥r0] be a nilpotent orbit of Type B, i.e.
∣{i∣ri=α}∣ is even for all even integers α.
Then LO is a special orbit if and only if its dual partition defines a nilpotent orbit of Type B or, equivalently,
r2l+1+r2l is even for all l≤k−1.*
•
Lg** of Type C:*** Let LO=[r2k+1≥r2k≥⋯≥r1] be a nilpotent orbit of Type C, i.e. ∣{i∣ri=α}∣ is even for all odd integers α.
Then LO is a special orbit if and only if its dual partition defines a nilpotent orbit of Type C or, equivalently,
r2l+1+r2l is even for all l≤k.*
•
Lg** of Type D:*** Let LO=[r2k+1≥r2k≥⋯≥r0] be a non-very even orbit of Type D, i.e. ∣{i∣ri=α}∣ is even for all even integers α. Then LO is a special orbit if and only if its dual partition defines a nilpotent orbit of Type C or, equivalently,
r2l+1+r2l is even for all l≤k.*
For any special orbit LO, the Lusztig’s quotient A(LO) is given by:
Lg** of Type B:*** Let LO=[r2k≥r2k−1≥⋯≥r0] be a special orbit. Separate all even rows (which must be of the form r2l+1=r2l=α), along with odd row pairs of the form r2l=r2l−1=β and get*
[TABLE]
•
Lg** of Type C:*** Let LO=[r2k+1≥r2k≥⋯≥r1] be a special orbit. Separate all odd rows (which must be of the form r2l+1=r2l=α), and even row pairs of the form r2l=r2l−1=β and get*
[TABLE]
•
Lg** of Type D:*** Let LO=[r2k+1≥r2k≥⋯≥r0] be a special, non-very even orbit. Separate all even rows (which must be of the form r2l+1=r2l=α), and all odd row pairs r2l=r2l−1=β and get*
[TABLE]
Then A(LO)=(Z/2Z)q, regardless of the number of αi’s and βj’s.
For any special orbit LO, consider
LO=LP∪LQ,
where LP is the orbit of the same type as LO without the αi’s,
and LQ:=⋃i=1x[αi,αi].
Note that LP is even with A(LP)≅A(LO) and the results in [BV3]
hold for LP. More precisely, the coordinates of
λO consists of:
•
Lg** of Type B:** integers coming from LP, half-integers coming from LQ.
•
Lg** of Type C:** half-integers coming from LP, integers coming from LQ.
•
Lg** of Type D:** integers coming from LP, half-integers coming from LQ.
Let Lg′ be the Lie subalgebra of Lg whose roots are
given by the roots α∨ in Lg satisfying
⟨α∨,λO⟩∈Z.
We will study integral special unipotent representations
Π(LO′), where LO′⊂Lg′=Lg1′+Lg2′
is an even orbit given by LO′=LP+LQ,
with
•
Lg** of Type B:** LP⊂Lg1′ is of Type B; LQ⊂Lg2′ is of Type D.
•
Lg** of Type C:** LP⊂Lg1′ is of Type C; LQ⊂Lg2′ is of Type C.
•
Lg** of Type D:** LP⊂Lg1′ is of Type D; LQ⊂Lg2′ is of Type D.
Note that A(O)=A(LO)=A(LP)=A(LO′)=A(O′).
We now study the unipotent representations Π(LP) and Π(LQ) individually.
For LQ, it has trivial Lusztig quotient, and the special unipotent representation attached to it is
Π(LQ)={IndGL(α1)×⋯×GL(αx)G2′(triv⊠⋯⊠triv)}.
Moreover, the (unique) left cell representation is VL(Q)=JAα1−1×⋯×Aαx−1W(G2′)(sgn⊠⋯⊠sgn).
On the other hand, suppose the left cell representation of P is VL(P)=x∈A(P)⨁σx′. Then
[TABLE]
For x∈A(O′)≅A(O), let
[TABLE]
One can use Equation (4.6.5) of [L1] to check that the right hand side is
non-zero and irreducible, and hence Step (VI) holds for all special O. More explicitly, Π(LO) is given by
[TABLE]
A.2. Exceptional Lie algebras
For exceptional special nilpotent orbits, we only focus on LO⊂Lg such that LO is not even. Since there are no such orbits in G2, we will only study exceptional Lie algebras of Type E and F.
In the following tables, we list all non-even special orbits LO and their Lusztig-Spaltenstein dual O.
In the third column, we get the subalgebra g′ of g determined by λO (Step (I)). Then λO determines an even orbit LO′⊂Lg′ (Step (II)), whose Lusztig-Spaltenstein dual O′ is recorded in the fourth column. Afterwards, we can use [L2] to compute VL(O′) (see Example A.3 below), and then we have VL(O)=⨁σ′∈VL(O′)JW(G′)W(G)(σ′) (Step(VI)), which is given in the second last column of the table, with σsp∈VL(O) always appears first in the list. And the last column records the degree deg(σsp)=deg(σsp′).
The computations below are carried out by LiE [LiE] and MATLAB.
A.2.1. Type F4
The results for F4 are as follows:
[TABLE]
A.2.2. Type E6
The results for E6 are as follows:
[TABLE]
A.2.3. Type E7
The results for E7 are as follows:
[TABLE]
A.2.4. Type E8
The results for E8 are as follows:
[TABLE]
Here is an example on how the above results are obtained:
Example A.3**.**
Let LO=D4(a1)+A1 be a nilpotent orbit of Type E8,
By calculating 21LH explicitly, one can check that
Lg′ is of Type E7+A1, and the coroots α∨
such that ⟨α∨,21LH⟩=0
forms a Lie subalgebra Ll′ of Type (A5+A1)+0.
Then LO′⊂Lg′ is the even orbit
[TABLE]
By looking at the tables of [Ca], one can check that
Lusztig-Spaltenstein dual of LO and LO′
are O=E8(a6) and O′=E7(a5)+[12]
respectively. Also, one can check from [CM] that
A(O)=A(O′)=S3.
We now study the left cell VL(O′):
Firstly, note that E7(a5)=IndE6E7(D4(a1)).
By [L2], the special piece attached to D4(a1)
is equal to {D4(a1),A3+A1,2A2+A1}, and their
Springer representation constitute the left cell
[TABLE]
(in fact,
VL(D4(a1)) can also be obtained
directly from (4.11.2) of [L1], but this
perspective is useful in determining left cells
for classical orbits with large Lusztig quotient).
Let [13], [21], [3] be the conjugacy classes
of S3, then by Proposition 4.14 of [BV3]
(which is valid since O′ is even),
[TABLE]
Using the notations of Step (III) in the first section,
σ[13]′=JE6+A1E7+A1(80s⊠sgn), σ[21]′=JE6+A1E7+A1(60s⊠sgn) and σ[3]′=JE6+A1E7+A1(10s⊠sgn).
By Proposition 3.18 of [BV2] and (4.13.3) of [L1], we have
[TABLE]
*Moreover, σsp=1400x is the special representation corresponding to the O under the Springer correspondence.
Therefore, this verifies Step (VI) and Table A.2.4 for this orbit.
*
We end the Appendix by mentioning the character formulas of the special unipotent
representations Π(LO) when O is equal to the three exceptional orbits, i.e.
A4+A1 in E7, A4+A1 and E6(a1)+A1
in E8.
From the Tables A.2.3 and A.2.4 above, one can check that for the O′⊂g′
corresponding to these O’s, A(O)≅A(O′)≅S2 and
[TABLE]
However, their truncated inductions are given by
[TABLE]
so VL(O)=σsp contains one irreducible representation only.
In other words, the second bullet point of Step (VI) does not hold for these orbits,
and one cannot express their character formulas in the form of Theorem III of [BV3].
Nevertheless, the character formulas of Π(LO)
can still be obtained by using Step (IV), and the number of representations
is equal to the number of irreducible representations of
A(O)≅A(O′).
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