Minimal Representations: Definition and Properties over Finite and Local Fields

Gordan Savin

July 8

Let G be a Chevalley group over a field F which is simply laced i.e. its root system is of type A, D or E. Denote by B a fixed Borel subgroup and W the associated Weyl group.

First consider when F is a finite field with q elements. Let 1 be the trivial representation of B. We then have the following isomorphisms:

End Ind_{B}^{G} (1) ≅ ℂ [B\G/B] ≅ ℂ [W] .

Associated with any character χ of W, there is a representation π_{χ} of G which is invariant on B. This association is a one-to-one correspondence.

The following are some examples of this correspondence.

1) If χ is the trivial character of W, then π_{χ} is the trivial representation of G.

2) Let χ is the sign on W, i.e., χ(w) = ±1 depending on whether w can be written as a product of even numbered simple transpositions. Then π_{χ} is the Steinberg representation on G.

3) Let χ be the reflection representation of W on the root space. Then π_{χ} is the smallest non-trivial representation of G. More precisely, let 𝔒_{min} be the smallest nontrivial nilpotent G-orbit. Then

dimΠ = q^{\frac{1}{2}dim(𝔒_{min})} + · · · .

We shall denote π_{χ} by Π and call it the minimal representation.

Now we discuss the p-adic case. Let I be the Iwahori subgroup and H = C_{c}^{∞} (I\G/I) be the Hecke algebra. In this case, Π again corresponds to the reflection representation of H (see Lusztig’s paper in Trans. AMS).

In what sense is Π minimal? Let (π, V) be any irreducible representation of the p-adic group G. Denote by K_{n} the n-th principal congruence subgroup. Then from the germ expansion of the character of (π, V):

Θ_{π} = ∑_{𝔒: nilpotent orbits}c_{𝔒}μ̂_{𝔒},

one obtains that (assuming the above expansion is normalized)

dimV^{K_{n}} = cq^{D} + ⋯ (lower order terms),

where

D = \frac{1}{2} max_{c_{𝔒}≠0} dim𝔒.

(Note that Mœglin and Waldspurger have shown that c_{𝔒} of leading orbits in the expansion are positive integers.) We have the following theorem:

Theorem 1 (Kazhdan-Savin) If π = Π, then D = \frac{1}{2}dim𝔒_{min} and c_{𝔒_{min}} = 1.

In particular, we see that the growth of K-types for Π is the smallest possible, justifying the name minimal.

We now present an example, over a finite field, how smallness of Π can be used to obtain results on dual pair correspondences. Assume that G = GL_{2n} over a finite field F of q elements. Let P = MN be the Siegel parabolic subgroup so that M ≅ GL_{n} × GL_{n}, N ≅ M_{n}.

Consider the restriction of Π to P (and N). Fix an additive character ψ : F → ℂ^{×}. Then we can obtain the additive characters of N as follows:

ψ_{A}(x) = ψ(Tr(Ax)), A ∈ M_{n}(F).

Remark that these are all the irreducible representations of N. Let π be any representation of GL_{2n}. Then

π|_{N} = ⊕π_{A}, π_{A} = {v ∈ π : π(x)v = π_{A}(x)v}.

Note that M permutes the summands. The M-orbits are parameterized by the rank of A. Let ω_{k} be the set of all n×n matrices of rank k, k = 1, ⋯, n. Then

|ω_{0}| = 1, |ω_{1}| = q^{2n−1} + ⋯, ⋯, |ω_{n}| = q^{n^{2}} + · · · .

If π_{A} ≠ 0 for some A in ω_{k}, then one can estimate

dimπ ≥ |ω_{k}|dimπ_{A}.

Applying the above argument to the minimal representation Π we see that

Π|_{N} = Π^{N} ⊕ ⨁_{A∈ω_{1}} Π_{A}, dimΠ_{A} = 1, if rankA = 1.

(We can do a similar thing in the p-adic case using the work of Mœglin and Waldspurger.) We summarize:

Theorem 2 We have

Π|_{P} = Π^{N} ⊕ ℂ[ω_{1}],

where the action of P = MN on f ∈ ℂ[ω_{1}] is given by

\{ \begin{matrix}Π (x) f(A) = ψ_{A}(x)f(A), x ∈ N; \\ Π (m) f(A) = f(mA), m ∈ M. \end{matrix}

Next, we define a dual pair, GL_{2} × GL_{n}, in GL_{2n} by

[ \begin{matrix}a & b \\ c & d \end{matrix} ] ⟼ [ \begin{matrix}aI_{n} & bI_{n} \\ cI_{n} & dI_{n} \end{matrix} ] , for [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ GL_{2};
g ⟼ [ \begin{matrix}g & 0 \\ 0 & g \end{matrix} ] , for g ∈ GL_{n}.

We consider the restriction

Π|_{GL_{2}×GL_{n}} = ⊕ (π ⊗ θ(π)) ,

where π is over all irreducible representations of GL_{2} (θ could be 0 or reducible). This can be further analyzed. But we turn to an application to symmetric spaces.

Let U = GL_{2} ∩ N which is isomorphic to F. Abusing notation, we define

ψ(( \begin{matrix}1 & u \\ 0 & 1 \end{matrix} )) = ψ(u) which is a character of U. Put

Π^{U,ψ} = {v ∈ Π | Π(u)v = ψ(u)v} .

Using the above restriction of Π to GL_{2} × GL_{n}, we see that

Π|_{GL_{2}×GL_{n}}^{U,ψ} = ⊕π^{U,ψ} ⊗ θ(π),

and since dimπ^{U,ψ} ≤ 1,

Π^{U,ψ} = ⊕θ(π),

where π runs over all generic representations of GL_{2}. On the other hand, by the previous theorem,

Π^{U,ψ} = (π^{N})^{U,ψ} ⊕ ℂ[ω_{1}]^{U,ψ}.

Of course, the first summand is 0 by the previous theorem. The second summand is equal to ℂ[Ω] where

Ω = {A ∈ ω_{1}| ψ_{A}|_{U} = ψ} = {A ∈ ω_{1}|TrA = 1} ,

which is a rank-one symmetric space GL_{n}/GL_{n−1}×GL_{1}. Finally, we conclude

Proposition 1

ℂ [GL_{n}/GL_{n−1} × GL_{1}] = ⊕θ(π),

where π runs over the generic representations of GL_{2}.

In particular we see that the decomposition of ℂ [GL_{n}/GL_{n−1} × GL_{1}] is a consequence of the dual pair correspondence. All rank-one symmetric spaces can be treated in this way. We can then give the following list. (A_{1} means the group of type A_{1}, etc.)

G ⊇ A_{1} × H dual pair.
Π^{U,ψ} = ℂ[H/K] = ⊕θ(π).

\begin{matrix}G & H & K \\ GL_{2n} & GL_{n} & GL_{n−1} × GL_{1} \\ Sp_{2n} & SO_{n} & SO_{n−1} \\ D_{2n} & Sp_{2n} & Sp_{2n} × Sp_{2n−2} \\ E_{7} & F_{4} & B_{4} \\ SO_{2n} & SO_{2n−6} & SO_{2n−4} \end{matrix}

For the second pair, see Howe’s paper in Journal of Functional Analysis for details.

Department of Mathematics, University of Utah, Salt Lake City, UT 84112

 savin@math.utah.edu