Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 38 (1997), No. 1, 145-148.

Correspondence between Representations of the Dual Reductive Pair O(n) and SL_{2}

Ilya Piatetski-Shapiro  Mina Teicher

Dept. of Mathematics Tel-Aviv University Ramat-Aviv, Israel Dept. of Math. & Comp. Sci. Bar-Ilan University Ramat-Gan 52900, Israel

Abstract. We state a connection between the Weil representation of the dual reductive pair O(n) and SL_{2}, and prove that they have eigenvectors in common.

1. Introduction

In this exposition we continue to study the Weil representation of Sp(2n, k) that we studied in “Commutativity of the Schur algebra”. There we were interested in the decomposition of its induced representation whereas here we are interested in the irreducible components of its restriction to the product of the dual reductive pair SL_{2} and O(n).

Dual reductive pairs are playing an important role in the study of representation theory. We shall prove here a result on the correspondence between representations of a dual reductive pair, based on Schur theory.

We recall the definition of a dual reductive pair.

Definition. A dual reductive pair for Sp(2n, k) is a pair G, H of reductive subgroups of Sp(2n, k). Each is the full centralizer of the other. (See Howe [1].)

2. Construction of the dual reductive pair O(n), SL_{2}

We are interested in the following examples of a dual reductive pair: Let k be a finite field. Let X = k^{n} with symmetric form ( , ) n ≥ 6, which the group O(n) = O(n, k) preserves. Let Y = k^{2} with some antisymmetric form ⟨ , ⟩, which SL_{2} = Sp(2, k) preserves. Look at Z = X ⊗ Y. Clearly dim Z = 2n and Z has an antisymmetric form induced by ( , ) and ⟨ , ⟩ as follows: (x_{1} ⊗ y_{1}, x_{2} ⊗ y_{2}) = (x_{1}, x_{2})⟨y_{1}, y_{2}⟩. The group Sp(2n) = Sp(2n, k)

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1997 Heldermann Verlag preserves its antisymmetric form. The action of O(n) on X induces an action of O(n) on Z which preserves ⟨ , ⟩. The action of SL_{2} on Y induces an action of SL_{2} on Z which preserves its antisymmetric form. Thus O(n) ↪ Sp(2n) and SL_{2} ↪ Sp(2n).

Denote the standard basis in X by e_{1}, …, e_{n} and in Y by y_{1}, y_{2}. Take the following basis in Z : e_{1} ⊗ y_{1}, …, e_{n} ⊗ y_{1}, e_{1} ⊗ y_{2}, …, e_{n} ⊗ y_{2}. With respect to this basis O(n) is

embedded by A → ( \begin{matrix}A & 0 \\ 0 & A \end{matrix} ) and SL_{2} by ( \begin{matrix}a & b \\ c & d \end{matrix} ) → ( \begin{matrix}aI_{n} & bI_{n} \\ cI_{n} & dI_{n} \end{matrix} ) . The groups O(n) and

SL_{2} are both reductive and they are each other’s centralizer. So they are a dual reductive pair.

Choose T = ( \begin{matrix}& & 1 \\ & … \\ 1 \end{matrix} ) to induce ( ) and J = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) to induce ⟨, ⟩. Then with

respect to the chosen basis of Z, ⟨, ⟩ is represented on Z by ( \begin{matrix}0 & T \\ −T & 0 \end{matrix} ) .

3. The Weil representation of O(n) × SL_{2}

Let (W_{ψ}, V) be a Weil representation of Sp(2n). (See Teicher [3] or Gerardin [1]; Thm. 2.4). To choose a model of W_{ψ} we have to choose a polarization. Take Z_{1} = X ⊗ y_{1}, Z_{2} = X ⊗ y_{2}. Then Z = Z_{1} + Z_{2}, Z_{1} and Z_{2} are maximal isotropic subspaces. In this case V = {ϕ : Z_{1} → ℂ}, Z_{1} ∼ X. Thus we can take V = {ϕ : X → ℂ}.

We shall consider X, when regarded as the domain of the definition for ϕ, as row vectors. We shall need the precise equation of O(n) acting on V :

W_{ϕ} ( \begin{matrix}T^{t}A^{−1}T & 0 \\ 0 & A \end{matrix} ) ϕ(x) = ϕ(xA), x ∈ k^{n}, A ∈ GL(n).

For A ∈ O(n), T^{t}A^{−1}T = A and A ↦ ( \begin{matrix}A & 0 \\ 0 & A \end{matrix} ) . So then:

W_{ϕ} ( \begin{matrix}A & 0 \\ 0 & A \end{matrix} ) ϕ(x) = ϕ(xA), x ∈ O(n).

W_{ϕ} |_{O(n)×SL_{2}}= ∑ τ × σ where σ are the irreducible representations of SL_{2}.

4. Construction of the character χ_{t}

Recall from part II of Teicher [4] the following notations:

L = Span(e_{2}, …, e_{n−1}), U the subgroup of O(n) characterized by u ∈ U ⇔ ∃a ∈ L

such that ue_{1} = e_{1}, uℓ = ℓ − (a, ℓ)e_{1} ∀ℓ ∈ L, ue_{n} = e_{1} + a − \frac{(a, a)}{2}e_{n}. Thus U →^{∼} L by

u → a. For m ∈ O(L), m^{−1}Um = U.

Looking at X as row vectors we then take e_{1} = (1, 0, …, 0), …, e_{n} = (0, …, 0, 1). Then (x, y) = xT^{t}y. The group O(n) then acts on X from the right and preserves ( , ) : g ∈ O(n) ⇒ gT^{t}g = T. Thus (xg, yg) = xgT^{t}(yg) = xgT^{t}g^{t}y = xT^{t}y = (x, y). We also have for u in U: e_{n}u = e_{n}, e_{1}u = e_{1} − aT − \frac{1}{2}(a, a)e_{n}.

Fix ψ, an additive character of k. Let t ∈ L. Let χ = χ_{t} be the following character of U: χ(u) = ψ((u, t)) when considering the isomorphism U →^{∼} L.

5. The main result

Theorem. Let τ ×σ be a component of (W_{ψ}, V) |_{O(n)×SL_{2}} . Let ϕ_{0} ∈ V be an eigenvector with respect to U, i.e., ∃t ∈ L such that for χ = χ_{t}, τ(u)ϕ_{0} = χ(u)ϕ_{0}. Assume that ϕ_{0} is supported on an orbit W of the right action of U on X, for W that satisfies (x, e_{n}) ≠ 0 for x ∈ W. Then ϕ_{0} is also an eigenvector with respect to M_{1} = Stab_{X}(O(L)).

Proof. Recall that we consider a function ϕ in V, ϕ : X → ℂ as a function on row vectors.

Take x_{1}, x_{2} ∈ W. W is an orbit under U, so there is u ∈ U such that x_{1}u = x_{2}. Therefore (x_{2}, e_{n}) = (x_{1}u, e_{n}) = (x_{1}u, e_{n}u) = (x_{1}, e_{n}) because e_{n}u = e_{n}. Since (x_{2}, x_{2}) = (x_{1}u, x_{1}u) = (x_{1}, x_{1}), we can write W ⊂ {x ∈ X | (x, e_{n}) = α, (x, x) = f}. We claim that equality actually holds. By the assumption α ≠ 0. Take x_{1}, x_{2} such that (x_{1}, e_{n}) = (x_{2}, e_{n}) = α ≠ 0 and (x_{1}, x_{1}) = (x_{2}, x_{2}) = f. A priori x_{i} = a_{i}e_{1} + ℓ_{i} + b_{i}e_{n} for ℓ_{i} ∈ L, i = 1, 2 where a_{i} = (x_{i}, e_{n}). Thus a_{1} = a_{2} = α ≠ 0. Let u_{i} ∈ U such that e_{1}u_{i} =

e_{1} − \frac{ℓ_{i}}{α} + \frac{(ℓ_{1}, ℓ_{1})}{2αα}e_{n}, i = 1, 2. Then x_{i}u_{i} = αe_{1} + c_{i}e_{n} for some c_{i} ∈ k, i = 1, 2, and

(x_{i}u_{i}, x_{i}u_{i}) = 2c_{i}α. On the other hand, (x_{i}u_{1}, x_{i}u_{1}) = (x_{i}, x_{k}) = f. Therefore c_{1} = c_{2} and x_{1}u_{1} = x_{2}u_{2}. Thus x_{2} = x_{1}u_{1}u_{2}^{−1} and x_{1} and x_{2} are in one orbit. This orbit must be W. So W = {x ∈ X | (x, e_{n}) = α, (x, x) = f}.

Take m ∈ M_{1} and look at τ(m)ϕ_{0} :

τ(m)ϕ_{0}(x) = W_{ψ} ( \begin{matrix}m & 0 \\ 0 & m \end{matrix} ) ϕ_{0}(x) = ϕ_{0}(xm). We claim that τ(m)ϕ_{0} is also an eigenvector

with respect to U :

(τ(u)(τ(m)ϕ_{0}))(x)=(τ(um)ϕ_{0})(x)=W_{ψ} (\begin{matrix}um & 0 \\ 0 & um \end{matrix}) ϕ_{0}(x)=ϕ_{0}(xum)=ϕ_{0}(xmm^{−1}um)
=χ(m^{−1}um)ϕ_{0}(xm)=χ(u)ϕ_{0}(xm)=χ(u)(τ(m)ϕ_{0})(x)

because m^{−1}um ∈ U ∀m ∈ O(L) and x(m^{−1}um) = x(u) ∀m ∈ M_{1}. Therefore τ(m)ϕ_{0} is also an eigenvector of U with respect to the same eigenvalue.

Take m ∈ M_{1} and x ∈ X. We have: (xm, xm) = (x, x) and m^{−1} fixes e_{n}. Thus (xm, e_{n}) = (x, e_{n}m^{−1}) = (x, e_{n}) and x ∈ W ⇔ xm ∈ W. Since ϕ_{0} was supported on W, and (τ(m)ϕ_{0})(x) = ϕ_{0}(xm) then τ(m)ϕ_{0} is also supported on W.

Therefore τ(m)ϕ_{0} is an eigenvector of U with respect to χ and supported on W. We claim that ϕ_{0} and τ(m)ϕ_{0} differ by a constant: Put W = Orbit(x_{0}), c_{0} = ϕ_{0}(x_{0}),

c_{1}(m) = (τ(m)ϕ_{0})(x_{0}). Take x ∈ W, x = (x_{0})u. Then (τ(m)ϕ_{0})(x) = (τ(m)ϕ_{0})((x_{0})u) =

x(u)τ(m)ϕ_{0})(x_{0}) = χ(u)c_{1}(m) = \frac{c_{1}(m)}{c_{0}}x(u)c_{0} = c_{m}x(u)ϕ_{0}(x_{0}) = c_{m}ϕ_{0}(x_{0}u) = c_{m}ϕ_{0}(x).

Thus τ(m)ϕ_{0} = c_{m}ϕ_{0}.

Since c_{m} is a character of M_{1} where M_{1} ∼ O(n − 3), n − 3 ≥ 3 then c_{m} = (det m)^{ε}. Thus τ(m)ϕ_{0} = (det m)_{ϕ_{0}}^{ε}.

Note that if ϕ is an eigenvector supported on {x | (x, x) = f} we can look at δ_{α}ϕ where δ_{α} is the charcteristic function of {x | (x, e_{n}) = α}, and get a vector that is supported on an orbit. Furthermore such vectors form a basis for the eigenvectors.

References

[1] Gérardin, P.: Weil’s representations associated to finite fields. J. of Algebra 46 (1977), 54–101.

[2] Howe, R.: θ series and invariant theory, part 1. Proceedings of Symposia in Pure Math. 33 (1979), 275–285.

[3] Teicher, M.: Commutativity of the Schur algebra. Beitr. Algebra und Geometrie 38 (1997), 135–143.

[4] Teicher, M.: Multiplicity one theorem for GS_{p}(k, 2n) and O(k, n). J. of Algebra 107 (1987), 436–465.

Received January 3, 1996