Small Representations of Finite Classical Groups

Abstract

Finite group theorists have established many formulas that express interesting properties of a finite group in terms of sums of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of representations of the group. In particular, the representations of small dimensions tend to contribute the largest terms to these sums, so a systematic knowledge of these small representations could lead to proofs of important conjectures which are currently out of reach. Despite the classification by Lusztig of the irreducible representations of finite groups of Lie type, it seems that this aspect remains obscure. In this note we develop a language which seems to be adequate for the description of the “small” representations of finite classical groups and puts in the forefront the notion of rank of a representation. We describe a method, the “eta correspondence”, to construct small representations, and we conjecture that our construction is exhaustive. We also give a strong estimate on the dimension of small representations in terms of their rank. For the sake of clarity, in this note we describe in detail only the case of the finite symplectic groups.

Appendices

Appendix 1: Proofs

1.1 Proof of Lemma 2.3.1

Proof

If Supp _N(ρ) = 0, then ρ _{| N} is a multiple of the trivial character. The Lemma now follows from the well-known fact that the N conjugates generate the group Sp [1]. □

1.2 Proof of Proposition 3.3.3

Proof

Consider the automorphism α _s: H → H given by α _s(v, z) = (sv, s ² z). This induces the equivalence of the oscillator representations ω _p si and $\omega _{p}si_{s^{2}}$. The fact that for a non-square $\varepsilon \in \mathbb{F}_{q}^{{\ast}}$, the representations ω _ψ and $\omega _{\psi _{\varepsilon }}$ are not isomorphic, can be verified using the realization given in Remark 3.3.3. This completes the proof of the proposition. □

1.3 Proof of Proposition 4.3.2

Proof

The proposition follows immediately from Eq. (23) in Sect. 4.2. □

Appendix 2: Proof of the Eta Correspondence Theorem

We give a proof of Theorem 4.3.3 that is an elementary application of the double commutant theorem [63].

2.1 The Double Commutant Theorem

We will use the following version:

Theorem A.1.1 (Double Commutant Theorem)

Let W be a finite dimensional vector space. Let $\mathcal{A},\mathcal{A}^{{\prime}}$ ⊂ End(W) be two sub-algebras, such that

• (1) The algebra $\mathcal{A}$ acts semi-simply on W. 

• (2) Each of $\mathcal{A}$ and $\mathcal{A}^{{\prime}}$ is the full commutant of the other in End(W). 

Then $\mathcal{A}^{{\prime}}$ acts semi-simply on W, and as a representation of  $\mathcal{A}\otimes \mathcal{A}^{{\prime}}$ we have

$$\displaystyle{ W =\bigoplus \limits _{i\in I}W_{i} \otimes W_{i}^{{\prime}}, }$$

where W _i are all the irreducible representations of $\mathcal{A}$ , and W _i ^′ are all the irreducible representations of $\mathcal{A}^{{\prime}}.$ In particular, we have a bijection between irreducible representations of $\mathcal{A}$ and $\mathcal{A}^{{\prime}},$ and moreover, every isotypic component for $\mathcal{A}$ is an irreducible representation of $\mathcal{A}\otimes \mathcal{A}^{{\prime}}.$

2.2 Preliminaries

Let us start with several preliminary steps. We work with the Schrödinger model of ω _{V ⊗U} appearing in Sect. 4.2. It is realized on the space

$$\displaystyle{ \mathcal{H} = L^{2}(Hom(X,U)), }$$

(31)

and there, the actions of an element A of the Siegel unipotent radical N ⊂ Sp, and an element r ∈ O _β, are given by Formulas (23) and (25), respectively. In particular, we have

Claim A.2.1

Every character appearing in the restriction of ω _{V ⊗U} to N is of the form $\psi _{\beta _{ T}}$ for some T ∈ Hom(X, U). Moreover, we have rank(β _T) = k iff T is onto.

For the rest of the section, we fix a transformation T: X ↠ U which is onto and consider the character subspace

$$\displaystyle{ \mathcal{H}^{\psi _{\beta _{T}}} =\{ f \in \mathcal{H};\ \omega _{V \otimes U}(A)f\ =\psi _{\beta _{ T}}(A)f\text{, }A \in N\}. }$$

We would like to have a better description of the space $\mathcal{H}^{\psi _{\beta _{T}}}$. The orthogonal group O _β acts naturally on Hom(X, U) and we denote by $\mathcal{O}_{T}$ the orbit of T under this action.

Proposition A.2.2

We have $\mathcal{H}^{\psi _{\beta _{T}}} = L^{2}(\mathcal{O}_{T})$ the space of functions on $\mathcal{O}_{T}.$

For a proof of Proposition A.2.2, see section “Proof of Proposition A.2.2”.

Note that, because T is onto, the action of O _β on $\mathcal{O}_{T}$ is free. In particular, we can identify $\mathcal{O}_{T}$ with O _β, and the Peter–Weyl theorem [51] for the regular representation implies

Corollary A.2.3

Under the action of O _β, the space $\mathcal{H}^{\psi _{\beta _{T}}}$ decomposes as

$$\displaystyle{ \mathcal{H}^{\psi _{\beta _{T}}} \simeq \bigoplus \limits _{\tau \in Irr(O_{ \beta })}\dim (\tau )\tau. }$$

(32)

We would like now to describe the commutant of O _β in $End(\mathcal{H}^{\psi _{\beta _{T}}}).$ Considering the group

$$\displaystyle{ G_{\beta _{T}} = Stab_{GL(X)}(\beta _{T}), }$$

of automorphisms of X that stabilize the form β _T, we obtain two commuting actions

$$\displaystyle{ O_{\beta } \curvearrowright \mathcal{H}^{\psi _{\beta _{T}}} \curvearrowleft G_{\beta _{ T}}. }$$

Moreover, we have

Proposition A.2.4

The groups O _β and $G_{\beta _{T}}$ generate each other’s commutant in $End(\mathcal{H}^{\psi _{\beta _{T}}}).$

For a proof of Proposition A.2.4, see section “Proof of Proposition A.2.4”.

2.3 Proof of Theorem 4.3.3

Proof

Write

$$\displaystyle{ \Theta (\tau ) \simeq \sum \eta _{i}(\tau ), }$$

for various irreducible representations η _i(τ) of Sp. Then

$$\displaystyle{ \Theta (\tau )^{\psi _{\beta _{T}}} \simeq \sum \eta _{i}(\tau )^{\psi _{\beta _{T}}}. }$$

(33)

In addition, $\Theta (\tau )^{\psi _{\beta _{T}}}$ is a $G_{\beta _{T}}$-module, and so is each $\eta _{i}(\tau )^{\psi _{\beta _{T}}}.$ Hence, Identity (33) gives a decomposition of $\Theta (\tau )^{\psi _{\beta _{T}}}$ into (not necessarily irreducible) submodules for $G_{\beta _{T}}$. But Proposition A.2.4 together with the Double Commutant Theorem says that $\Theta (\tau )^{\psi _{\beta _{T}}}$ is irreducible as a $G_{\beta _{T}}$-module. Therefore, exactly one of the $\eta _{i}(\tau )^{\psi _{\beta _{T}}}$ will be non-zero, and it defines an irreducible representation of $G_{\beta _{T}}$, which has dimension equal to dim(τ), by Eq. (32). To conclude, there exists a unique irreducible sub-representation η(τ) < $\Theta (\tau )$ of rank k and type $\mathcal{O}_{\beta _{T}},$ the multiplicity of the orbit $\mathcal{O}_{\beta }$ in η(τ)_{| N} is dim(τ), and finally, the Double Commutant Theorem implies that for $\tau \not\cong \tau ^{{\prime}}$ in Irr(O _β), we have $\eta (\tau ) \not\cong \eta (\tau ^{{\prime}}).$ This completes the proof of Theorem 4.3.3. □

2.4 Proofs

2.4.1 Proof of Proposition A.2.2

Proof

Using the delta basis {δ _T; T ∈ Hom(X, U)}, we can verify Claim A.2.2, by showing that if $\beta _{T^{{\prime}}} =\beta _{T}$ then there exists r ∈ O _β such that T ^′ = r ∘ T. Indeed, let r be the composition

$$\displaystyle{ U\widetilde{\longrightarrow }X/rad(\beta _{T})\widetilde{\longrightarrow }U, }$$

where the first and second isomorphisms are these induced by T ^′, and T, respectively, and rad(β _T) is the radical of β _T. This completes the proof of Proposition A.2.2. □

2.4.2 Proof of Proposition A.2.4

Proof

To verify this assertion, note that we have a short exact sequence

$$\displaystyle{ 1 \rightarrow N_{k,n-k} \rightarrow G_{\beta _{T}} \rightarrow O(X/rad(\beta _{T})) \times GL(Z) \rightarrow 1, }$$

(34)

where Z = ker(T) = rad(β _T), O(X∕rad(β _T)) is the orthogonal group of X∕rad(β _T), and N _k, n−k is the appropriate unipotent group. The group O(X∕rad(β _T)) acts simply transitively on the orbit $\mathcal{O}_{T}$, as does the group O _β, and these two actions commute with each other. If we use the map r ↦ r ⁻¹ ∘ T to identify O _β with $\mathcal{O}_{T}$, then the action of O _β becomes the action of O _β on itself by left translation, and the action of O(X∕rad(β _T)) can be identified with the action of O _β on itself by right translation. By the Peter-Weyl Theorem [51], we conclude that the groups O(X∕rad(β _T)) and O _β generate mutual commutants in the operators on $L^{2}(\mathcal{O}_{T}) \simeq \mathcal{H}^{\psi _{\beta _{T}}}$. A fortiori the groups $G_{\beta _{T}}$ and O _β generate mutual commutants on $L^{2}(\mathcal{O}_{T})$. This completes the proof of Proposition A.2.4. □