Principal series for general linear groups over finite commutative rings

Tyrone Crisp; Ehud Meir; Uri Onn

arXiv:1704.05575·math.RT·August 20, 2020

Principal series for general linear groups over finite commutative rings

Tyrone Crisp, Ehud Meir, Uri Onn

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TL;DR

This paper constructs a family of representations for the general linear group over any finite commutative ring, extending the principal series concept beyond finite fields.

Contribution

It introduces a new class of representations for $ ext{GL}_n(R)$ over finite commutative rings, generalizing principal series representations from finite fields.

Findings

01

Representations mirror intertwining properties of classical principal series.

02

Extension of principal series to general finite commutative rings.

03

Framework applicable to a broad class of rings.

Abstract

We construct, for any finite commutative ring $R$ , a family of representations of the general linear group $GL_{n} (R)$ whose intertwining properties mirror those of the principal series for $GL_{n}$ over a finite field.

Equations76

i : Rep (L) \to Rep (G) X \mapsto C [G] e_{U} e_{V} \otimes_{C [L]} X

i : Rep (L) \to Rep (G) X \mapsto C [G] e_{U} e_{V} \otimes_{C [L]} X

r : Rep (G) \to Rep (L) Y \mapsto e_{U} e_{V} C [G] \otimes_{C [G]} Y,

r : Rep (G) \to Rep (L) Y \mapsto e_{U} e_{V} C [G] \otimes_{C [G]} Y,

0 \to G_{r} ≅ (M_{n} (k), +) \to G (R) \to G (R / m^{r}) \to 0,

0 \to G_{r} ≅ (M_{n} (k), +) \to G (R) \to G (R / m^{r}) \to 0,

dim_{C} (Hom_{G} (i χ, i σ)) = # {w \in W ∣ w^{*} χ = σ} .

dim_{C} (Hom_{G} (i χ, i σ)) = # {w \in W ∣ w^{*} χ = σ} .

(U \cap U^{w}) \times (U \cap V^{w}) \to U and (V \cap U^{w}) \times (V \cap V^{w}) \to V

(U \cap U^{w}) \times (U \cap V^{w}) \to U and (V \cap U^{w}) \times (V \cap V^{w}) \to V

B w_{0} B \cap t^{- 1} B r w_{0} B = \emptyset.

B w_{0} B \cap t^{- 1} B r w_{0} B = \emptyset.

B w_{0} B \cap t^{- 1} B r w_{0} B \subseteq (B w_{0} B \cap t_{1}^{- 1} B sr w_{0} B) ⊔ (B w_{0} B \cap t_{1}^{- 1} B r w_{0} B) .

B w_{0} B \cap t^{- 1} B r w_{0} B \subseteq (B w_{0} B \cap t_{1}^{- 1} B sr w_{0} B) ⊔ (B w_{0} B \cap t_{1}^{- 1} B r w_{0} B) .

φ_{w} : e_{U^{w}} e_{V^{w}} C [G] x \mapsto e_{V} x e_{V} e_{U} C [G]

φ_{w} : e_{U^{w}} e_{V^{w}} C [G] x \mapsto e_{V} x e_{V} e_{U} C [G]

e_{V} e_{U^{w}} e_{V^{w}} C [G] = e_{V} e_{U} e_{V^{w}} C [G] \subseteq e_{V} e_{U} C [G]

e_{V} e_{U^{w}} e_{V^{w}} C [G] = e_{V} e_{U} e_{V^{w}} C [G] \subseteq e_{V} e_{U} C [G]

w^{- 1} z w e_{U^{w}} e_{V^{w}} (e_{V} e_{U^{w}} e_{V^{w}} f) = z^{w} (e_{U^{w}} e_{V^{w}})^{2} f = e_{U^{w}} e_{V^{w}} f

w^{- 1} z w e_{U^{w}} e_{V^{w}} (e_{V} e_{U^{w}} e_{V^{w}} f) = z^{w} (e_{U^{w}} e_{V^{w}})^{2} f = e_{U^{w}} e_{V^{w}} f

Φ : C [w L] \to e_{U} e_{V} C [G_{w}] e_{U} e_{V} w l \mapsto e_{U} e_{V} w l e_{U} e_{V}

Φ : C [w L] \to e_{U} e_{V} C [G_{w}] e_{U} e_{V} w l \mapsto e_{U} e_{V} w l e_{U} e_{V}

Φ (w h) = e_{U} e_{V} e_{U^{w^{- 1}}} e_{V^{w^{- 1}}} w h .

Φ (w h) = e_{U} e_{V} e_{U^{w^{- 1}}} e_{V^{w^{- 1}}} w h .

e_{U^{w^{- 1}}} e_{V^{w^{- 1}}} C [G] x \mapsto e_{V} x e_{V} e_{U} C [G]

e_{U^{w^{- 1}}} e_{V^{w^{- 1}}} C [G] x \mapsto e_{V} x e_{V} e_{U} C [G]

e_{V} e_{U} C [G] x \mapsto e_{U} x e_{U} e_{V} C [G]

e_{V} e_{U} C [G] x \mapsto e_{U} x e_{U} e_{V} C [G]

w h \mapsto e_{U^{w^{- 1}}} e_{V^{w^{- 1}}} w h = w e_{U} h e_{V}

w h \mapsto e_{U^{w^{- 1}}} e_{V^{w^{- 1}}} w h = w e_{U} h e_{V}

g x = x (x^{- 1} g x) \in V^{w} G_{0} = V^{w} (V_{0}^{w} L_{0} U_{0}^{w}) = V^{w} L_{0} U_{0}^{w}

g x = x (x^{- 1} g x) \in V^{w} G_{0} = V^{w} (V_{0}^{w} L_{0} U_{0}^{w}) = V^{w} L_{0} U_{0}^{w}

e_{V} w l g e_{U} e_{V}

e_{V} w l g e_{U} e_{V}

= e_{V} w l g (∣ U \cap V^{w} ∣^{- 1} x \in U \cap V^{w} \sum x) e_{U^{w}} e_{V \cap V^{w}}

= ∣ U \cap V^{w} ∣^{- 1} x \in U \cap V^{w} \sum e_{V} w l α (x) β (x) γ (x) e_{U^{w}} e_{V \cap V^{w}} .

e_{V} w l g e_{U} e_{V}

e_{V} w l g e_{U} e_{V}

= e_{V} w h e_{U \cap V^{w}} e_{U \cap U^{w}} e_{V \cap U^{w}} e_{V \cap V^{w}} = e_{V} w h e_{U} e_{V},

⟨ ζ^{2} e_{U^{y}} (e_{U} e_{V} t h_{t} e_{U} e_{V}) e_{U} e_{V} r h_{r} e_{U} e_{V} ⟩ = ⟨ ζ^{2} e_{U^{y}} e_{U} e_{V} e_{U^{y}} e_{V^{y}} t h_{t} r h_{r} e_{U^{r}} e_{V^{r}} e_{U} e_{V} ⟩

⟨ ζ^{2} e_{U^{y}} (e_{U} e_{V} t h_{t} e_{U} e_{V}) e_{U} e_{V} r h_{r} e_{U} e_{V} ⟩ = ⟨ ζ^{2} e_{U^{y}} e_{U} e_{V} e_{U^{y}} e_{V^{y}} t h_{t} r h_{r} e_{U^{r}} e_{V^{r}} e_{U} e_{V} ⟩

= ⟨ ζ^{2} e_{U^{y}} e_{U} e_{V} e_{U^{y}} e_{V^{y}} t h_{t} e_{V} e_{U} e_{V^{r}} e_{U^{r}} h_{r}^{*} r ⟩ = ⟨ ζ^{2} e_{U^{y}} e_{U} e_{V} e_{U^{y}} e_{V^{y}} e_{U^{y}} e_{V^{r y}} e_{U^{r y}} t h_{t} h_{r}^{*} r ⟩

= ⟨ ζ^{2} e_{U^{y}} e_{U} e_{V^{y}} e_{U^{y}} e_{V^{y}} e_{U^{y}} e_{V^{y}} e_{U^{r y}} t h_{t} h_{r}^{*} r ⟩ = ⟨ ζ^{2} (e_{U^{y}} e_{V^{y}})^{3} e_{U^{r y}} t h_{t} h_{r}^{*} r ⟩

= ⟨ e_{U^{y}} e_{V^{y}} e_{U^{r y}} t h_{t} h_{r}^{*} ∣ r ⟩ = ⟨ t e_{U} h_{t} h_{r}^{*} e_{V} e_{U^{r}} ∣ r ⟩ = ⟨ e_{U} h_{t} h_{r}^{*} e_{V} ∣ t^{- 1} e_{U} r ⟩ = 0.

0

0

\displaystyle=\Big{\langle}\zeta^{2}e_{U^{y}}e_{U}e_{V}th_{t}e_{U}e_{V}\mid e_{U}e_{V}th_{t}e_{U}e_{V}\Big{\rangle}

\displaystyle=\Big{\langle}\zeta e_{U^{y}}e_{U}e_{V}th_{t}e_{U}e_{V}\mid\zeta e_{U^{y}}e_{U}e_{V}th_{t}e_{U}e_{V}\Big{\rangle},

e_{U} e_{V} C [G] e_{U} e_{V} = w \in W \sum e_{U} e_{V} C [G_{w}] e_{U} e_{V} .

e_{U} e_{V} C [G] e_{U} e_{V} = w \in W \sum e_{U} e_{V} C [G_{w}] e_{U} e_{V} .

C [W ⋉ L] = w \in W ⨁ C [w L] h \mapsto e_{U} e_{V} h e_{U} e_{V} e_{U} e_{V} C [G] e_{U} e_{V}

C [W ⋉ L] = w \in W ⨁ C [w L] h \mapsto e_{U} e_{V} h e_{U} e_{V} e_{U} e_{V} C [G] e_{U} e_{V}

χ_{1} \otimes \dots \otimes χ_{n} : diag (r_{1}, \dots, r_{n}) \mapsto χ_{1} (r_{1}) \dots χ_{n} (r_{n})

χ_{1} \otimes \dots \otimes χ_{n} : diag (r_{1}, \dots, r_{n}) \mapsto χ_{1} (r_{1}) \dots χ_{n} (r_{n})

i χ ≅ C [G] e_{U} e_{V} \otimes_{C [L]} C [L] e_{χ} ≅ C [G] e_{U} e_{V} e_{χ} = C [G] z e_{U} e_{V} e_{χ}

i χ ≅ C [G] e_{U} e_{V} \otimes_{C [L]} C [L] e_{χ} ≅ C [G] e_{U} e_{V} e_{χ} = C [G] z e_{U} e_{V} e_{χ}

e_{L (R)} e_{U (R)} e_{V (R)} C [G (R)] e_{U (R)} e_{V (R)} e_{L (R)} ⟶ e_{L (k)} e_{U (k)} e_{V (k)} C [G (k)] e_{U (k)} e_{V (k)} e_{L (k)} .

e_{L (R)} e_{U (R)} e_{V (R)} C [G (R)] e_{U (R)} e_{V (R)} e_{L (R)} ⟶ e_{L (k)} e_{U (k)} e_{V (k)} C [G (k)] e_{U (k)} e_{V (k)} e_{L (k)} .

C [G] \to C [G], g \mapsto τ (det g) g

C [G] \to C [G], g \mapsto τ (det g) g

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Full text

Principal series for general linear groups over finite

commutative rings

Tyrone Crisp

Department of Mathematics and Statistics, University of Maine, Orono, ME 04469-5752, USA

[email protected]

,

Ehud Meir

Institute of Mathematics, University of Aberdeen, Fraser Noble Building, Aberdeen AB24 3UE, UK

[email protected]

and

Uri Onn

Mathematical Sciences Institute, The Australian National University, Canberra, Australia

[email protected]

(Date: Last updated )

Abstract.

We construct, for any finite commutative ring $R$ , a family of representations of the general linear group $\operatorname{GL}_{n}(R)$ whose intertwining properties mirror those of the principal series for $\operatorname{GL}_{n}$ over a finite field.

Key words and phrases:

Harish-Chandra induction, principal series, general linear groups, finite commutative rings

2010 Mathematics Subject Classification:

Primary 20G05; Secondary 20C33, 20C15, 20C05.

1. Introduction

Among the irreducible, complex representations of reductive groups over finite fields, the simplest to construct and to classify are the principal series: those obtained by Harish-Chandra induction from a minimal Levi subgroup; see, for instance, [HK80]. In this paper we use a generalisation of Harish-Chandra induction to construct a ‘principal series’ of representations of the group $\operatorname{GL}_{n}(R)$ , where $R$ is any finite commutative ring with identity. Our main results assert that the well-known intertwining relations among the principal series for $\operatorname{GL}_{n}$ over a finite field also hold for the representations that we construct.

The study of the principal series for reductive groups over finite fields can be viewed as the first step in the program to understand all irreducible complex representations of such groups in terms of what Harish-Chandra called the ‘philosophy of cusp forms’ [HC70, Spr70b]. This program has met with considerable success. The basic ideas appear already in Green’s determination [Gre55] of the irreducible characters of $\operatorname{GL}_{n}(\Bbbk)$ , where $\Bbbk$ is a finite field, and these ideas have since been developed and generalised to a very great extent; see [DM91] for an overview.

The theory for groups over finite rings is in a far less advanced state. Most efforts so far have been directed toward groups over principal ideal rings: see [Hil93, Hil95a, Hil95b, Hil94, Lus04, Lus17, Onn08, Sta11, CS17, SS17, KOS18, Sta17]. By contrast, the results presented below are valid for all finite rings, and they depend on the algebraic properties of the base ring in only a very limited way. For instance, we give a uniform construction of a family of irreducible representations of $\operatorname{GL}_{n}(R)$ for all finite local rings $R$ , and to our knowledge these are the first results obtained in this degree of generality.

The present paper is part of a project whose aim is to extend the philosophy of cusp forms to reductive groups over finite rings. Our construction, which is a special case of a general induction procedure developed in [CMO19], extends in a natural way to produce more general ‘Harish-Chandra series’. The analysis of the intertwining properties of these more general series seems, however, to be substantially more involved than the results for the principal series presented here. See [CMO19, Section 5] and [CMO20] for some partial results in this more general setting.

Notation and definitions.

Let $R$ be a finite commutative ring with $1$ . Let $G=\operatorname{GL}_{n}(R)$ , let $L\cong(R^{\times})^{n}$ be the subgroup of diagonal matrices in $G$ , and let $U$ and $V$ be the upper-unipotent subgroup and the lower-unipotent subgroup, respectively, in $G$ . We write $G(R)$ , $L(R)$ , etc., when it is necessary to specify $R$ .

The ring $R$ decomposes as a direct product of local rings: $R\cong R_{1}\times\cdots\times R_{m}$ , and this decomposition is unique up to permuting the factors [McD74, Theorem VI.2]. There is a corresponding decomposition $G(R)\cong G(R_{1})\times\cdots\times G(R_{m})$ , and similarly for $L$ , $U$ , and $V$ . If $R$ is a local ring then we let $N(R)$ be the subgroup of monomial matrices in $G(R)$ , that is, products of permutation matrices with diagonal matrices. If $R$ is not local then we define $N(R)=N(R_{1})\times\cdots\times N(R_{m})$ , where the $R_{i}$ are the local factors of $R$ as above. Let $W(R)=N(R)/L(R)$ . It will be convenient to realise $W(R)$ as a subgroup of $G(R)$ , as follows: if $R$ is local, then we identify $W(R)$ with the group of permutation matrices; and in the general case we identify $W(R)$ with the product of the permutation subgroups in $G(R)\cong G(R_{1})\times\cdots\times G(R_{m})$ . Note that following Lemma 4, we will be able to assume without loss of generality that $R$ is a local ring.

If $\chi:L\to\operatorname{GL}(X)$ is a representation of $L$ on a complex vector space $X$ , and if $w\in W$ , then we let $w^{*}\chi$ denote the representation $\chi\circ{\operatorname{Ad}_{w}^{-1}}:L\to\operatorname{GL}(X)$ . We let $W_{\chi}=\{w\in W\ |\ w^{*}\chi\cong\chi\}$ .

For each subgroup $H\subseteq G$ we let $e_{H}$ denote the idempotent in the complex group ring $\mathbb{C}[G]$ corresponding to the trivial character of $H$ : $e_{H}=|H|^{-1}\sum_{h\in H}h$ . Since $L$ normalises $U$ and $V$ , the idempotents $e_{U}$ and $e_{V}$ commute with $\mathbb{C}[L]$ inside $\mathbb{C}[G]$ .

We consider the functors

[TABLE]

where $\operatorname{Rep}(G)$ denotes the category of complex representations, identified in the usual way with the category of left $\mathbb{C}[G]$ -modules. This is a special case of the construction defined in [CMO19, Section 2], which generalises a definition due to Dat [Dat09]. The functors $\operatorname{i}$ and $\operatorname{r}$ are two-sided adjoints to one another; see [CMO19, Theorem 2.15] for a proof of this and other basic properties.

Definition.

Let us say that an irreducible representation of $G$ is in the principal series if it is isomorphic to a subrepresentation of $\operatorname{i}\chi$ for some representation $\chi$ of $L$ .

Example.

If $R$ is a field, then the map $\mathbb{C}[G]e_{U}\xrightarrow{f\mapsto fe_{V}}\mathbb{C}[G]e_{V}$ is known to be an isomorphism of $\mathbb{C}[G]$ - $\mathbb{C}[L]$ bimodules; see [HL94, Theorem 2.4]. It follows that the functors $\operatorname{i}$ and $\operatorname{r}$ are naturally isomorphic to the familiar functors of Harish-Chandra induction and restriction, i.e. the functors of tensor product with the bimodules $\mathbb{C}[G]e_{U}$ and $e_{U}\mathbb{C}[G]$ , respectively. The same is not true if $R$ is not a product of fields. Revisiting an example from [CMO19], let $1_{L}$ denote the trivial representation of $L$ . Then $\mathbb{C}[G]e_{U}\otimes_{\mathbb{C}[L]}1_{L}\cong\mathbb{C}[G/LU]$ , with $G$ acting by permutations of $G/LU$ . The number of distinct irreducible subrepresentations in $\mathbb{C}[G/LU]$ depends in a delicate way on the underlying ring $R$ ; for instance, for $G=\operatorname{GL}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ with $k\geq 2$ and $n\geq 3$ , this number depends on both $p$ and $k$ [OPV06]. By contrast, it follows from Theorem 2 below that for any $R$ the number of distinct irreducible subrepresentations of $\operatorname{i}1_{L}$ is equal to $P(n)^{m}$ , where $G=\operatorname{GL}_{n}(R)$ , $P$ is the partition function, and $m$ is the number of maximal ideals in $R$ .

Example.

Suppose that $R$ is a finite discrete valuation ring, with maximal ideal $\mathfrak{m}$ and residue field $\Bbbk$ , and let $r$ be the largest integer such that $\mathfrak{m}^{r}\neq 0$ . Reduction modulo $\mathfrak{m}^{r}$ gives rise to a group extension

[TABLE]

which one can use to study the representations of $G(R)$ via Clifford theory; see [Hil93], for example. In [Hil95a], Hill identified a class of representations that are particularly amenable to this approach: an irreducible representation $\pi$ of $G(R)$ is called regular if its restriction to $G_{r}$ contains a character whose stabiliser under the adjoint action of $G(\Bbbk)$ has minimal dimension. Explicit constructions of all such representations are given in [SS17, KOS18].

An application of [CMO19, Theorem 3.4] gives the following criterion for regularity of the induced representations $\operatorname{i}\chi$ : if $\chi$ is an irreducible representation of $L(R)$ , then $\operatorname{i}\chi$ is regular if and only if the restriction of $\chi$ to the subgroup $L(R)\cap G_{r}\cong\Bbbk^{n}$ has trivial stabiliser under the permutation action of $S_{n}$ . Moreover, the representations $\operatorname{i}\chi$ , for $\chi$ satisfying the above condition, account for all of the regular representations associated to the split semisimple classes in $M_{n}(\Bbbk)$ .

For $n=2$ , all of the principal series representations of $G(R)=\operatorname{GL}_{2}(R)$ can be described in terms of regular representations, as follows. Let $\chi:L\to\mathbb{C}^{\times}$ be an irreducible representation of $L$ . If $\operatorname{i}\chi$ is irreducible, then there is a character $\tau:R^{\times}\to\mathbb{C}^{\times}$ , an integer $k$ , and a regular representation $\pi$ of $G(R/\mathfrak{m}^{k})$ associated to a split semisimple class in $M_{n}(\Bbbk)$ such that $\operatorname{i}\chi$ is isomorphic to the representation $(\tau\circ\det)\otimes\pi$ , where $\pi$ is pulled back to a representation of $G(R)$ . If $\operatorname{i}\chi$ is not irreducible, then there is a character $\tau:R^{\times}\to\mathbb{C}^{\times}$ such that $\operatorname{i}\chi$ is isomorphic to the representation $(\tau\circ\det)\otimes(1_{G}\oplus\operatorname{St})$ , where $1_{G}$ is the trivial representation, and $\operatorname{St}$ is the Steinberg representation of $G(\Bbbk)$ pulled back to $G(R)$ . (These assertions follow easily from Theorem 3 and Lemma 15, below, and from [CMO19, Theorem 3.4].) For $n\geq 3$ the relationship between the principal series and the regular representations becomes more complicated.

2. Main results

We will show that the following well-known properties of the Harish-Chandra functors are shared by the functors $\operatorname{i}$ and $\operatorname{r}$ for $R$ an arbitrary finite commutative ring.

Theorem 1.

There is a natural isomorphism $\operatorname{r}\operatorname{i}\cong\bigoplus_{w\in W}w^{*}$ of functors on $\operatorname{Rep}(L)$ . Consequently, if $\chi$ and $\sigma$ are irreducible representations of $L$ , then

[TABLE]

When $\sigma=\chi$ , we have the following more precise statement:

Theorem 2.

For each irreducible representation $\chi$ of $L$ one has $\operatorname{End}_{G}(\operatorname{i}\chi)\cong\mathbb{C}[W_{\chi}]$ as algebras.

Theorems 1 and 2 readily imply the following combinatorial formula for the number of principal series representations. Following [And08], we let $P_{k}(n)$ denote the number of multipartitions of $n$ with $k$ parts: i.e., the number of $k$ -tuples $(\lambda^{(1)},\ldots,\lambda^{(k)})$ , where each $\lambda^{(i)}$ is a partition of some non-negative integer $n_{i}$ , and $\sum_{i}n_{i}=n$ .

Corollary 3.

If $R$ is isomorphic to a product $R_{1}\times\cdots\times R_{m}$ of finite local rings, and for each $j$ we set $k_{j}=|R_{j}^{\times}|$ , then the principal series of $\operatorname{GL}_{n}(R)$ contains precisely $\prod_{j}P_{k_{j}}(n)$ distinct isomorphism classes of irreducible representations.

Remarks.

$\square$

In the case where $R$ is a field, Theorems 1 and 2 are essentially due to Green [Gre55]; see [Ste51] for the case $\chi=1_{L}$ , and see [Spr70a] for an exposition. Both of these results have been generalised to arbitrary Harish-Chandra series for arbitrary reductive groups: see [HC70] and [HL80], respectively. 2. $\square$

Theorems 1 and 2 can be extended, using [CMO19, Theorem 2.15(5)], to the setting of smooth representations of the profinite groups $G(\mathcal{O})$ , where $\mathcal{O}$ is the ring of integers in a nonarchimedean local field.

3. Proofs

The first step in the proof of the main results is to reduce to the case of local rings.

Lemma 4.

If Theorems 1 and 2 and Corollary 3 are true for all finite commutative local rings, then they are true for all finite commutative rings.

Proof.

Let $R$ be a finite commutative ring, and write $R$ as a product of local rings $R_{1}\times\cdots\times R_{m}$ . All of the groups and the representation categories in Theorems 1 and 2 and in Corollary 3 then decompose into products accordingly: $G(R)\cong G(R_{1})\times\cdots\times G(R_{m})$ , $\operatorname{Rep}(G(R))\cong\operatorname{Rep}(G(R_{1}))\times\cdots\times\operatorname{Rep}(G(R_{m}))$ , and so on. The bimodule $\mathbb{C}[G(R)]e_{U(R)}e_{V(R)}$ decomposes as the tensor product of the bimodules $\mathbb{C}[G(R_{j})]e_{U(R_{j})}e_{V(R_{j})}$ , and likewise for $e_{U(R)}e_{V(R)}\mathbb{C}[G(R)]$ , so the functors $\operatorname{i}$ and $\operatorname{r}$ are compatible with the above decompositions. By definition, the group $W$ also decomposes compatibly. Thus Theorems 1 and 2 and Corollary 3 over $R$ follow immediately from the corresponding results over the local factors $R_{j}$ . ∎

Assume from now on that $\boldsymbol{R}$ is a local ring. Let $\mathfrak{m}$ denote the maximal ideal of $R$ , and let $\Bbbk$ denote the residue field $R/\mathfrak{m}$ . Recall that $W\cong S_{n}$ is then the group of permutation matrices in $G$ . We write $\ell$ for the word-length function on $W$ with respect to the standard generating set $S=\{(1\ 2),\dots,(n-1\ n)\}$ .

The following proposition collects the group-theoretical ingredients of the proof of Theorem 1.

Proposition 5.

(a)

The multiplication map $U\times L\times V\to G$ is injective. 2. (b)

The reduction-mod- $\mathfrak{m}$ * map $G(R)\to G(\Bbbk)$ is surjective.* 3. (c)

For each subgroup $H$ of $G$ , let $H_{0}$ denote the intersection of $H$ with the kernel $G_{0}$ of the above reduction homomorphism. Then the multiplication map $U_{0}\times L_{0}\times V_{0}\to G_{0}$ is a bijection, and the same is true for any ordering of the three factors. 4. (d)

For each $w\in W$ the multiplication maps

[TABLE]

are bijections, where $U^{w}=w^{-1}Uw$ , etc. 5. (e)

$G$ * is the disjoint union $G=\bigsqcup_{w\in W}G_{w}$ , where $G_{w}=V{wL}UG_{0}$ .* 6. (f)

For each $r,t\in W$ with $\ell(t)\leq\ell(r)$ and $t\neq r$ one has $ULV\cap t^{-1}Ur=\emptyset$ .

Proof.

Parts (a), (b), (c) and (d) are well-known and easily verified. Part (e) follows immediately from the Bruhat decomposition of $G(\Bbbk)$ [CR87, (65.4)].

In part (f) we may assume without loss of generality that $R$ is a field, since $ULV\cap t^{-1}Ur$ is empty if its reduction modulo $\mathfrak{m}$ is empty. Let ${w_{0}}$ denote the longest element $(1,2,\ldots,n)\mapsto(n,\ldots,2,1)$ of $W$ , and write $B$ for the upper-triangular subgroup $LU$ of $G$ . We will show that under the stated assumptions on $t$ and $r$ one has

[TABLE]

Since $ULV{w_{0}}=UL{w_{0}}U=B{w_{0}}B$ , while $t^{-1}Ur{w_{0}}\subseteq t^{-1}Br{w_{0}}B$ , we see that (6) implies that $ULV\cap t^{-1}Ur=\emptyset$ .

To prove (6) we recall (from, e.g., [CR87, (65.10)]) that $(B,N,W,S)$ is a BN-pair in $G$ (note that we are assuming $R$ to be a field). This means, among other things, that $sBx\subseteq BsxB\sqcup BxB$ for all $s\in S$ and all $x\in W$ ; and that $BxB\cap ByB=\emptyset$ for all $x\neq y\in W$ . The proof of (6) is by induction on $\ell(t)$ . If $\ell(t)=0$ , so that $t=1$ , then $B{w_{0}}B\cap Br{w_{0}}B=\emptyset$ unless $r=1$ . For the inductive step, write $t=st_{1}$ with $s\in S$ and $\ell(t_{1})=\ell(t)-1$ . We then have

[TABLE]

We have $\ell(t_{1})=\ell(t)-1\leq\ell(r)-1\leq\ell(sr)$ , and $st_{1}\neq r\implies t_{1}\neq sr$ , so the first term in the union is empty by induction. We have $\ell(t_{1})<\ell(t)\leq\ell(r)$ , and in particular $t_{1}\neq r$ , so the second term in the union is also empty. This proves (6). ∎

We equip $\mathbb{C}[G]$ with the Hermitian inner product $\langle\ |\ \rangle$ for which the group elements $g\in G$ constitute an orthonormal basis; and with the conjugate-linear involution $*$ defined on basis elements by $g^{*}=g^{-1}$ . The two structures are related by the identity $\langle abc|d\rangle=\langle b|a^{*}{d}c^{*}\rangle$ for all $a,b,c,d\in\mathbb{C}[G]$ . An element $a\in\mathbb{C}[G]$ is called self-adjoint if $a=a^{*}$ .

Lemma 7.

Let $A$ denote the unital subalgebra of $\mathbb{C}[G]$ generated by the idempotents $e_{U}$ and $e_{V}$ . There is a self-adjoint, invertible element $z$ in the centre of $A$ such that $z(e_{U}e_{V})^{2}=e_{U}e_{V}$ and $z(e_{V}e_{U})^{2}=e_{V}e_{U}$ .

Proof.

This is true for any pair of orthogonal projections on a finite-dimensional Hilbert space: see [Hal69, Theorem 2], for example. ∎

Remark.

If $R$ is a field then [HL94, Theorem 2.4] implies that there is a unique element $z$ as in Lemma 7. This is not the case over a general ring.

Lemma 8.

For each $w\in W$ we have $e_{V}e_{U^{w}}e_{V^{w}}=e_{V}e_{U}e_{V^{w}}$ .

Proof.

It is clear that $e_{V}=e_{V}e_{(V\cap U^{w})}$ and similarly that $e_{U}=e_{(U\cap U^{w})}e_{U}$ . Proposition 5(d) gives $e_{(V\cap U^{w})}e_{(U\cap U^{w})}=e_{U^{w}}$ , and it follows that $e_{V}e_{U}=e_{V}e_{U^{w}}e_{U}$ . The same reasoning gives $e_{U^{w}}e_{V^{w}}=e_{U^{w}}e_{U}e_{V^{w}}$ , and so $e_{V}e_{U}e_{V^{w}}=e_{V}e_{U^{w}}e_{U}e_{V^{w}}=e_{V}e_{U^{w}}e_{V^{w}}$ . ∎

Lemma 9.

For each $w\in W$ the map

[TABLE]

is an isomorphism of $\mathbb{C}[L]$ - $\mathbb{C}[G]$ bimodules.

Proof.

The following argument is taken from [Dat09, Lemme 2.9]. The map $\varphi_{w}$ is well-defined, because

[TABLE]

by Lemma 8. The map $\varphi_{w}$ is injective, because for each $f\in\mathbb{C}[G]$ we have

[TABLE]

where $z$ is as in Lemma 7, and in the first equality we used that $V=(V\cap V^{w})(V\cap U^{w})$ . The domain and target of $\varphi_{w}$ are isomorphic as vector spaces: indeed, $e_{V}e_{U}\mathbb{C}[G]=w_{0}we_{U^{w}}e_{V^{w}}\mathbb{C}[G]$ , where $w_{0}$ is the longest element of $W$ . Since $\varphi_{w}$ is injective it is thus also an isomorphism. ∎

For each subset $K\subseteq G$ , we let $\mathbb{C}[K]$ denote the vector subspace of $\mathbb{C}[G]$ spanned by $K$ .

Proposition 10.

For each $w\in W$ the map

[TABLE]

is an isomorphism of $\mathbb{C}[L]$ -bimodules.

Proof.

$\Phi$ is clearly a bimodule map. Let us show that it is injective. For $h\in\mathbb{C}[L]$ we have

[TABLE]

The maps

[TABLE]

and

[TABLE]

are isomorphisms by Lemma 9, so we are left to prove that the map

[TABLE]

is injective on $\mathbb{C}[wL]$ . It is, because Proposition 5(a) implies that the cosets $UlV$ are all disjoint as $l$ ranges over $L$ . Thus $\Phi$ is injective.

To prove that $\Phi$ is surjective, first note that $G_{w}=VwLG_{0}U$ because $G_{0}$ is normal in $G$ . Since $e_{V}v=e_{V}$ and $ue_{U}=e_{U}$ for all $v\in V$ and $u\in U$ , we find that $e_{U}e_{V}\mathbb{C}[G_{w}]e_{U}e_{V}$ is spanned by elements of the form $e_{U}e_{V}wlge_{U}e_{V}$ , where $l\in L$ and $g\in G_{0}$ . We will show that each element of this form is in the image of $\Phi$ .

For each $x\in V^{w}$ we have

[TABLE]

by Proposition 5(c). Let $\alpha:V^{w}\to V^{w}$ , $\beta:V^{w}\to L_{0}$ and $\gamma:V^{w}\to U_{0}^{w}$ be the (unique) functions satisfying $gx=\alpha(x)\beta(x)\gamma(x)$ for all $x\in V^{w}$ . Writing $e_{U}=e_{U\cap V^{w}}e_{U\cap U^{w}}$ and $e_{V}=e_{V\cap U^{w}}e_{V\cap V^{w}}$ , we then have

[TABLE]

Since $\gamma(x)\in U^{w}$ we have $\gamma(x)e_{U^{w}}=e_{U^{w}}$ for each $x\in U\cap V^{w}$ . Since $\alpha(x)\in V^{w}$ we have $wl\alpha(x)l^{-1}w^{-1}\in V$ , and consequently $e_{V}wl\alpha(x)=e_{V}wl$ for each $x$ . Continuing the computation with the space-saving notation $h=|U\cap V^{w}|^{-1}\sum_{x\in U\cap V^{w}}l\beta(x)\in\mathbb{C}[L]$ , we find that

[TABLE]

and so $e_{U}e_{V}wlve_{U}e_{V}=\Phi(wh)$ . ∎

Proposition 11.

The set $\{e_{U}e_{V}wle_{U}e_{V}\in\mathbb{C}[G]\ |\ w\in W,\ l\in L\}$ is linearly independent.

Proof.

We know from Proposition 10 that for each $w\in W$ the set $\{e_{U}e_{V}wle_{U}e_{V}\ |\ l\in L\}$ is linearly independent. We must show that for different choices of $w$ these sets are independent from one another.

Suppose we had elements $h_{w}\in\mathbb{C}[L]$ , not all zero, with $\sum_{w\in W}e_{U}e_{V}wh_{w}e_{U}e_{V}=0$ . Let $t\in W$ be an element of minimal length such that $h_{t}$ is nonzero. To compactify the notation we shall write $y=t^{-1}$ .

Let $z$ be as in Lemma 7, and write $\zeta=y^{-1}zy$ . Thus $\zeta$ is a self-adjoint, invertible element of $\mathbb{C}[G]$ which commutes with $e_{U^{y}}$ and $e_{V^{y}}$ and which satisfies $\zeta(e_{U^{y}}e_{V^{y}})^{2}=e_{U^{y}}e_{V^{y}}$ . For each $r\in W$ with $r\neq t$ such that $h_{r}\neq 0$ we have

[TABLE]

Here we have repeatedly used the equality $\langle abc|d\rangle=\langle b|a^{*}dc^{*}\rangle$ ; in the fourth step we used Lemma 8 to replace $e_{U}e_{V}e_{U^{y}}$ with $e_{U}e_{V^{y}}e_{U^{y}}$ and to replace $e_{U^{y}}e_{V^{ry}}e_{U^{ry}}$ with $e_{U^{y}}e_{V^{y}}e_{U^{ry}}$ ; in the fifth step we used Proposition 5(d) to write $e_{U^{y}}e_{U}e_{V^{y}}=e_{U^{y}}e_{V^{y}}$ ; and in the final equality we used Proposition 5(f), which applies because of the minimality of $\ell(t)$ , and which implies that the functions $e_{U}h_{t}h_{r}^{*}e_{V}$ and $t^{-1}e_{U}r$ are supported on disjoint subsets of $G$ and are therefore orthogonal.

It follows from this that

[TABLE]

where the last equality holds because $\zeta$ is self-adjoint, $e_{U^{y}}$ is a self-adjoint idempotent, and $\zeta$ and $e_{U^{y}}$ commute. Thus $\zeta e_{U^{y}}e_{U}e_{V}th_{t}e_{U}e_{V}=0$ . Since $\zeta$ is invertible, and left multiplication by $e_{U^{y}}$ is injective on $e_{U}e_{V}\mathbb{C}[G]$ (Lemma 9), we conclude that $e_{U}e_{V}th_{t}e_{U}e_{V}=0$ . By Proposition 10 this implies that $h_{t}=0$ , contradicting our choice of $t$ and completing the proof of the proposition. ∎

Proof of Theorem 1.

The functor $\operatorname{r}\operatorname{i}$ is naturally isomorphic to the functor of tensor product (over $\mathbb{C}[L]$ ) with the $\mathbb{C}[L]$ -bimodule $e_{U}e_{V}\mathbb{C}[G]e_{U}e_{V}$ , while the functor $\bigoplus_{w\in W}w^{*}$ is naturally isomorphic to the tensor product with the bimodule $\mathbb{C}[W\ltimes L]$ . Since $G=\bigsqcup G_{w}$ we have

[TABLE]

Proposition 10 thus implies that the $\mathbb{C}[L]$ -bimodule map

[TABLE]

is surjective. Proposition 11 implies that this map is injective, so it is an isomorphism of bimodules, and induces a natural isomorphism of functors $\operatorname{r}\operatorname{i}\cong\bigoplus w^{*}$ . The formula for the intertwining number follows from this isomorphism and from the fact that $\operatorname{i}$ and $\operatorname{r}$ are adjoints. ∎

We now turn to the proof of Theorem 2. Every irreducible representation $\chi$ of the abelian group $L\cong(R^{\times})^{n}$ has the form

[TABLE]

where each $\chi_{i}$ is a linear character $R^{\times}\to\mathbb{C}^{\times}$ . For each such $\chi$ we let $e_{\chi}=|L|^{-1}\sum_{l\in L}\chi(l)^{-1}l$ be the corresponding primitive central idempotent in $\mathbb{C}[L]$ .

Lemma 12.

The algebra $\operatorname{End}_{G}(\operatorname{i}\chi)$ is isomorphic to the subalgebra $e_{\chi}e_{U}e_{V}\mathbb{C}[G]e_{U}e_{V}e_{\chi}$ of $\mathbb{C}[G]$ .

Proof.

We have

[TABLE]

where $z$ is as in Lemma 7. Since $ze_{U}e_{V}$ and $e_{\chi}$ are commuting idempotents in $\mathbb{C}[G]$ , their product $E=ze_{U}e_{V}e_{\chi}$ is an idempotent and we have $\operatorname{End}_{G}(\mathbb{C}[G]E)\cong\left(E\mathbb{C}[G]E\right)^{\operatorname{opp}}$ via the action of $E\mathbb{C}[G]E$ on $\mathbb{C}[G]E$ by right multiplication. Now $E\mathbb{C}[G]E$ is a finite-dimensional complex semisimple algebra, so it is isomorphic to its opposite, and we have $E\mathbb{C}[G]E=e_{\chi}e_{U}e_{V}\mathbb{C}[G]e_{U}e_{V}e_{\chi}$ . ∎

Lemma 13.

For the trivial representation $1_{L}$ of $L$ we have $\operatorname{End}_{G}(\operatorname{i}1_{L})\cong\mathbb{C}[W]$ as algebras.

Proof.

First suppose that $R$ is a field, so that the functor $\operatorname{i}$ is isomorphic to the functor of Harish-Chandra induction. Then, as we noted above, $\operatorname{i}1_{L}$ is isomorphic to the permutation representation on $\mathbb{C}[G/LU]$ , and the isomorphism $\operatorname{End}_{G}(\operatorname{i}1_{L})\cong\mathbb{C}[W]$ is a special case of well-known results of Iwahori-Matsumoto and Tits (see [CR87, §68] for an exposition).

Now let $R$ be a local ring with residue field $\Bbbk$ . The quotient map $R\to\Bbbk$ induces a surjective map of algebras

[TABLE]

Theorem 1 implies that the domain of (14) is isomorphic as a vector space to $\mathbb{C}[W]$ , while we have just seen that the range of (14) is isomorphic as an algebra to $\mathbb{C}[W]$ . Since (14) is surjective, it is an algebra isomorphism. ∎

Remark.

The isomorphism in Lemma 13 is not canonical. One can trace through the various maps appearing in the proof to construct a set of Iwahori-Hecke generators of $e_{L}e_{U}e_{V}\mathbb{C}[G]e_{U}e_{V}e_{L}$ , although this will depend on the choice of an element $z$ as in Lemma 7.

Lemma 15.

Let $\chi=\chi_{1}\otimes\cdots\otimes\chi_{n}$ be an irreducible representation of $L$ , let $\tau:R^{\times}\to\mathbb{C}^{\times}$ be a character of $R^{\times}$ , and let $\chi^{\prime}=\tau\chi_{1}\otimes\cdots\otimes\tau\chi_{2}$ . Then $\operatorname{i}\chi^{\prime}\cong(\tau\circ\det)\otimes\operatorname{i}\chi$ .

Proof.

The algebra automorphism

[TABLE]

sends $e_{\chi^{\prime}}$ to $e_{\chi}$ , and fixes $e_{U}$ and $e_{V}$ . Thus (16) induces an isomorphism of $\mathbb{C}[G]$ -modules

[TABLE]

Lemma 17.

If $\chi=\chi_{1}^{n}$ is a tensor-multiple of a single character of $R^{\times}$ , then $\operatorname{End}_{G}(\operatorname{i}\chi)\cong\operatorname{End}_{G}(\operatorname{i}1_{L})$ as algebras.

Proof.

Lemma 15 ensures that $\operatorname{i}\chi\cong(\chi_{1}\circ\det)\otimes\operatorname{i}1_{L}$ . ∎

Lemma 18.

For each $w\in W$ there is a natural isomorphism of functors $\operatorname{i}\circ w^{*}\cong\operatorname{i}$ .

Proof.

The functor $\operatorname{i}\circ w^{*}$ is given by tensor product with the $\mathbb{C}[G]$ - $\mathbb{C}[L]$ bimodule $\mathbb{C}[G]e_{U}e_{V}w=\mathbb{C}[G]e_{U^{w}}e_{V^{w}}$ , while the functor $\operatorname{i}$ is given by tensor product with $\mathbb{C}[G]e_{U}e_{V}$ . These two bimodules are isomorphic, by Lemma 9. ∎

Lemma 18 implies that in order to compute the intertwining algebra $\operatorname{End}_{G}\left(\operatorname{i}\chi\right)$ for an arbitrary character $\chi$ of $L$ we may permute the factors $\chi_{i}$ so that $\chi$ takes the form

[TABLE]

(The exponents indicate tensor powers.) We then have $W_{\chi}\cong S_{n_{1}}\times\cdots\times S_{n_{k}}$ .

In the next lemma we shall consider general linear groups of different sizes, and we shall accordingly embellish the notation with subscripts to indicate the size of the matrices involved: so, for example, $L_{a}$ denotes the diagonal subgroup in $G_{a}=\operatorname{GL}_{a}(R)$ , and $\operatorname{i}_{a}$ is a functor from $\operatorname{Rep}(L_{a})$ to $\operatorname{Rep}(G_{a})$ .

Lemma 20.

If $\chi$ is as in (19) then $\operatorname{End}_{G_{n}}(\operatorname{i}_{n}\chi)\cong\bigotimes_{j=1}^{k}\operatorname{End}_{G_{n_{j}}}\big{(}\operatorname{i}_{n_{j}}(\chi_{j}^{n_{j}})\big{)}$ as algebras.

Proof.

Let us write $L^{\prime}$ for the block-diagonal subgroup $G_{n_{1}}\times\cdots\times G_{n_{k}}\subseteq G_{n}$ , which contains as subgroups the groups $U^{\prime}=U_{n_{1}}\times\cdots\times U_{n_{k}}$ and $V^{\prime}=V_{n_{1}}\times\cdots\times V_{n_{k}}$ . Let $U^{\prime\prime}$ be the subgroup of block-upper-unipotent matrices

[TABLE]

and let $V^{\prime\prime}=(U^{\prime\prime})^{\mathrm{t}}$ be the corresponding group of block-lower-unipotent matrices.

Let $\operatorname{i}^{\prime}:\operatorname{Rep}(L^{\prime})\to\operatorname{Rep}(G_{n})$ be the functor of tensor product with the $\mathbb{C}[G_{n}]$ - $\mathbb{C}[L^{\prime}]$ bimodule $\mathbb{C}[G_{n}]e_{U^{\prime\prime}}e_{V^{\prime\prime}}$ . The semidirect product decompositions $U=U_{n}=U^{\prime}\ltimes U^{\prime\prime}$ and $V=V_{n}=V^{\prime}\ltimes V^{\prime\prime}$ give equalities $e_{U}=e_{U^{\prime}}e_{U^{\prime\prime}}$ and $e_{V}=e_{V^{\prime}}e_{V^{\prime\prime}}$ , and hence an isomorphism of $\mathbb{C}[G_{n}]$ - $\mathbb{C}[L_{n}]$ bimodules

[TABLE]

It follows that

[TABLE]

Since $\operatorname{i}^{\prime}$ is a functor, we obtain from this isomorphism a map of algebras

[TABLE]

Now, the $\mathbb{C}[L^{\prime}]$ -bimodule map

[TABLE]

is injective, because the multiplication map $L^{\prime}\times U^{\prime\prime}\times V^{\prime\prime}\to G_{n}$ is one-to-one. It follows from this that the identity functor on $\operatorname{Rep}(L^{\prime})$ is a subfunctor of $\operatorname{Res}^{G_{n}}_{L^{\prime}}\circ\operatorname{i}^{\prime}$ . Thus $\operatorname{i}^{\prime}$ is a faithful functor, and in particular the map (21) is injective. Since the domain and the range of this map have the same dimension as complex vector spaces, by Theorem 1, we conclude that (21) is an algebra isomorphism. ∎

Proof of Theorem 2.

Lemma 18 allows us to assume that $\chi$ has the form (19), and in this case we have algebra isomorphisms

[TABLE]

Proof of Corollary 3.

Choose an ordering $\{\chi_{1},\ldots,\chi_{k}\}$ of the character group $\widehat{R^{\times}}$ . Lemma 18 and the intertwining number formula in Theorem 1 imply that for each principal series representation $\pi$ of $\operatorname{GL}_{n}(R)$ there is a unique $k$ -tuple of non-negative integers $n_{1},\ldots,n_{k}$ having $\sum_{i}n_{i}=n$ , such that $\pi$ embeds in $\operatorname{i}(\chi_{1}^{n_{1}}\otimes\cdots\otimes\chi_{k}^{n_{k}})$ .

Theorem 2 implies that the number of distinct irreducible subrepresentations of $\operatorname{i}(\chi_{1}^{n_{1}}\otimes\cdots\otimes\chi_{k}^{n_{k}})$ is equal to the number of distinct irreducible representations of $S_{n_{1}}\times\cdots\times S_{n_{k}}$ . The latter number is equal to the number of $k$ -tuples $(\lambda^{(1)},\ldots,\lambda^{(k)})$ , where each $\lambda^{(i)}$ is a partition of $n_{i}$ . Allowing the exponents $n_{i}$ to vary shows that the total number of principal series representations is equal to $P_{k}(n)$ , as claimed. ∎

Acknowledgments

The first and second authors were partly supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92). The first author was also supported by fellowships from the Max Planck Institute for Mathematics in Bonn, and from the Radboud Excellence Initiative at Radboud University Nijmegen. The second author was also supported by the Research Training Group 1670 “Mathematics Inspired by String Theory and Quantum Field Theory.” The third author acknowledges the support of the Israel Science Foundation and of the Australian Research Council.

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TL;DR

Contribution

Findings

Abstract

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Principal series for general linear groups over finite

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

Notation and definitions.

Definition**.**

Example**.**

Example**.**

2. Main results

Theorem 1**.**

Theorem 2**.**

Corollary 3**.**

Remarks**.**

3. Proofs

Lemma 4**.**

Proof.

Proposition 5**.**

Proof.

Lemma 7**.**

Proof.

Remark**.**

Lemma 8**.**

Proof.

Lemma 9**.**

Proof.

Proposition 10**.**

Proof.

Proposition 11**.**

Proof.

Proof of Theorem 1.

Lemma 12**.**

Proof.

Lemma 13**.**

Proof.

Remark**.**

Lemma 15**.**

Proof.

Lemma 17**.**

Proof.

Lemma 18**.**

Proof.

Lemma 20**.**

Proof.

Proof of Theorem 2.

Proof of Corollary 3.

Acknowledgments

Definition.

Example.

Example.

Theorem 1.

Theorem 2.

Corollary 3.

Remarks.

Lemma 4.

Proposition 5.

Lemma 7.

Remark.

Lemma 8.

Lemma 9.

Proposition 10.

Proposition 11.

Lemma 12.

Lemma 13.

Remark.

Lemma 15.

Lemma 17.

Lemma 18.

Lemma 20.