EXCEPTIONAL THETA CORRESPONDENCE OVER FINITE FIELDS

WEE TECK GAN

1. Lusztig Series

Let k be the finite field of characteristic p with q elements, and \overline{k} a fixed algebraic closure of k. Let G be a connected reductive group over \overline{k}, and assume for simplicity that G has connected center, and that p is a good prime for G. Suppose that F : G ⟶ G is a Frobenius map such that G^{F} := G(k) is a finite group of Lie type. In this section, we briefly recall the theory of Deligne and Lusztig on the complex representation theory of G(k).

Recall that associated to the pair (G, F), there is a dual pair (G^{∗}, F^{∗}) [Ca]. In our case, G^{∗} has simply-connected derived group. Let Irr(G(k)) denote the set of equivalences classes of irreducible representations of G(k) (over ℂ or \overline{ℚ}_{l}, with l ≠ p). Then it is a result of Deligne and Lusztig that:

Irr(G(k)) = ⋃_{(s^{∗})⊂G^{∗}} Irr(G(k), s^{∗})  (1.1)

a disjoint union over the F^{∗}-stable semi-simple classes (s^{∗}) in the dual group G^{∗}. Each set Irr(G(k), s^{∗}) is called a geometric conjugacy class or a Lusztig series.

Example: The set Irr(G, 1) is the set of unipotent characters, which include the trivial representation and the Steinberg representation.

It was shown further by Lusztig that there is a bijection (almost canonical)

Irr(G(k), s^{∗}) ⟶ Irr(C_{G^{∗}}(s^{∗}), 1)  (1.2)

Hence, in each Lusztig series, there is a character corresponding to the Steinberg character of C_{G^{∗}}(s^{∗}) under the above bijection (though the bijection above is not always canonical, the preimage of the Steinberg representation is). This is the unique generic character in the series. Recall that an irreducible character of G(k) is said to be generic if it is a component of the Gelfand-Graev representation, which is multiplicity-free, and the generic representations are characterized by the fact that their dimensions have the form:

q^{N} + (lower powers of q) where N is the number of positive roots of G. Hence we see that generic characters can be parametrized by F^{∗}-stable semi-simple classes in the dual group G^{∗}. We shall denote by χ_{s^{∗}}^{G} the generic character corresponding to the class (s^{∗}).

Date: 25 July 1997.

2. Reflection Representation

Now assume that G is simple and split over k with rank l. In this section, we focus attention on a very special representation Π of G(k), which is called the reflection representation. Π can be characterized by one of the following two statements:

(i) Π is the unipotent principal series representation whose space of Borelfixed vectors is isomorphic, as a representation of the Weyl group W of G, to the reflection representation ξ of W.

(ii) Let P be any parabolic subgroup, and suppose the Levi factor of P has semi-simple rank r. Then,

⟨Π, Ind_{P}^{G}1⟩ = l − r

If, further, G is simply-laced, it was shown by Lusztig that, at least for p a good prime for G, we have:

Π = \frac{1}{|W |} ∑_{w∈W} ξ(w)R_{w}  (2.1)

Here, R_{w} := R_{T_{w},1} is a Deligne-Lusztig character. From this, one deduce that, for a semi-simple element s ∈ G(k),

T r(s|Π) = T r(F |M(s))  (2.2)

where M(s) is the motive of C_{G}(s)^{0} [Gr].

Example: dim(Π) = ∑_{i} q^{e_{i}}, where the e_{i}’s are the exponents of W. Note that if e_{1} is the largest exponent, then 2e_{1} is the dimension of the minimal nilpotent orbit in the Lie algebra of G. Hence we are justified to call Π the minimal representation of G(k).

3. Dual Pair Correspondence

In view of the discussion in the previous section, we can use Π to obtain correspondences of representatons for dual pairs. This is the finite field analogue of the results of Magaard and Savin [M-S] over p-adic fields. We shall concentrate only on the pair

G_{2} × P GL_{3} ⊂ E_{6}

Here all three groups are of adjoint type. We refer the reader to [G] for more details.

Let π be a generic representation of P GL_{3}. Then we let Θ(π) be the set of irreducible representations π′ of G_{2}, counted with multiplicity, such that π ⊗ π′ is a component of Π|_{G_{2}×P GL_{3}}. Let Θ_{g}(π) ⊂ Θ(π) be the subset of generic representations. Then we have:

Proposition 3.1. Assume that p ≥ 5. Then

Θ_{g}(χ_{s^{∗}}^{P GL_{3}}) = {χ_{i(s^{∗})}^{G_{2}}}

where

i : SL_{3} ⟶ G_{2}

is the natural inclusion of dual groups.

Remark: Note that P GL_{3}(k) has q^{2} generic characters, each of which has degree (q^{3} + ...). Also, each generic character of G_{2}(k) has degree (q^{6} + ...). Thus the proposition accounts for a subspace of Π of dimension (q^{11} + ...). Since the dimension of Π is q^{11} + q^{8} + q^{7} + q^{5} + q^{4} + q, the proposition accounts for a large piece of Π.

Let us give a sketch of the proof of this proposition, which essentially consists of the following three steps:

(i) Whittaker Vectors. Let U_{G_{2}} be the unipotent radical of a Borel subgroup of G_{2}, and let φ be a character of U_{G_{2}} in general position. Then we have the following proposition.

Proposition 3.2.

Π_{U,φ} = Gelfand-Graev representation of P GL_{3}(k) .

As a corollary, we have:

Corollary 3.3. If π is a generic representation of P GL_{3}(k), then Θ_{g}(π) is a singleton set.

(ii) Jacquet Modules. In view of (i), it remains to chack that the single element in Θ_{g}(χ_{s^{∗}}^{P GL_{3}}) is indeed χ_{i(s^{∗})}^{G_{2}}. We first do this for non-cuspidal representations. Then this reduces to a computation of Jacquet functors. In this case, the computation in [M-S] works over a finite field as well. As an example, let us describe one of the results in [M-S].

Let P = MN be a maximal parabolic subgroup of E_{6}, with M of type D_{5} and N abelian. Then one sees easily, for reasons of dimension, that:

Π|_{P} = Π_{N} ⊕ ℂ[Ω]

where Π_{N} = 1⊕Π(M) (Π(M) is the minimal representation of M), and Ω is the minimal non-trivial M-orbit on \overline{N}. Moreover, if ψ is a fixed non-trivial character of k, and ⟨·, ·⟩ is a non-degenerate G-invariant form on the Lie algebra of G, for example the Killing form, then we can identify \overline{N} and the dual N^{∨} of N as follows:

ψ_{x}(n) = ψ(⟨n, x⟩), n ∈ N, x ∈ \overline{N}

Now for f ∈ ℂ[Ω], then

Π(n)f(x) = ψ_{x}(n)f(x), n ∈ N
Π(m)f(x) = f(m^{−1}(x)), m ∈ M

Note that (G_{2} ×P GL_{3})∩P = G_{2} ×LU, where LU is a maximal parabolic of P GL_{3}, so that L = GL_{2}. Using the above description of Π|_{P}, it was shown by Magaard and Savin that:

Proposition 3.4. As a G_{2} × L-module,

ℂ[Ω]_{𝕌} ≅ 𝕀⋉_{𝕃×ℙ_{⊭}}^{𝕃×𝔾_{⊭}}ℂ[𝔾𝕃_{⊭}] ⊕ 𝕀⋉_{𝔹×ℙ_{⊮}}^{𝕃×𝔾_{⊭}}ℂ[𝔾𝕃_{⊮}]

Here B is the Borel subgroup of L, P_{2} is the Heisenberg parabolic of G_{2}, and P_{1} is the other maximal parabolic of G_{2}.

In the above proposition, only the first summand on the right hand side can contain generic representation of G_{2}(k). From this, it is easy to check that the correspondence for non-cuspidal representations is indeed functorial with respect to the natural inclusion of dual groups.

(iii) Base Change/Shintani Descent. It remains to verify the proposition for cuspidal characters. For these, we shall use base change to reduce to the case of non-cuspidal representations. Let us recall briefly the notion of base change.

Let k_{m} be the extension of k of degree m. Then there is a norm map N_{m} : G(k_{m}) → G(k) which maps the F -conjugacy classes of G(k_{m}) bijectively onto the conjugacy classes of G(k). Hence, if χ is an F -class function on G(k_{m}), then the function Sh_{m}(χ)(N_{m}(g)) := χ(g) is a class function on G(k), called the Shintani descent of χ. Note that Sh_{m} defines a vector space isomorphism between the space of F -class function on G(k_{m}) and the space of class function on G(k). More importantly, it is an isometry for the natural inner products on the two spaces:

⟨Sh_{m}(χ_{1}), Sh_{m}(χ_{2})⟩_{G(k)} = ⟨χ_{1}, χ_{2}⟩_{G(k_{m})}

In particular, if χ is an F -stable representation of G(k_{m}), it extends to a representation of G(k_{m}) ⋊ ⟨F ⟩ (non-uniquely of course), and the function

g ↦ χ(F g)

is an F -class function on G(k_{m}), and so we have Sh_{m}(χ(F ·)), which of course may not be the character of a representation of G(k). However we have the following results of Digne-Michel and Digne [D]:

Proposition 3.5. (i) Let Π_{m} be the reflection representation of G(k_{m}). Then there is an extension of Π_{m} to G(k_{m}) ⋊ ⟨F ⟩ such that:

Sh_{m}(Π_{m}(F ·)) = Π

(ii) If p is a good prime for G and (m, p) = 1, then there is an extension of the Deligne-Lusztig charcater R_{T_{m},θ_{m}}, where T_{m} = T (k_{m}) and θ_{m} = θ ∘ N_{m}, such that:

Sh_{m}(R_{T_{m},θ_{m}}(F ·)) = R_{T (k),θ}

Here, θ is a character of T (k).

In our case, we need to assume p ≥ 5 to apply this proposition. Using these results and the fact the Sh_{m} is an isometry, one reduces to the case of non-cuspidal representations of the groups over k_{m}, and the results in (ii) apply. This completes the proof of the proposition.

Remarks: (i) We note that the computations in (i) and (ii) above works over a p-adic field as well. However, we do not know how to adapt (iii) to the p-adic case, or whether it is possible to do so.

(ii) Essentially the same computations give us similar results for the dual pairs G_{2} × P U_{3} ⊂ ^{2}E_{6} and G_{2} × P GSp_{6} ⊂ E_{7}. Since every semi-simple element of G_{2} is contained in either a SL_{3} or a SU_{3}, we see that every generic character of G_{2}(k) is lifted from either P GL_{3}(k) or P U_{3}(k) by the theta correspondence.

4. Classical Case

We would like to conclude with a short survey of the results for the classical dual pairs over finite fields. Let W be the Weil representation of Sp_{2n}, and let H_{1} × H_{2} ⊂ Sp_{2n}(k) be a classical dual pair. Hence, (H_{1}, H_{2}) = (GL_{m}, GL_{m′}), (U_{m}, U_{m′}) or (Sp_{2m}, O_{m′}). In [Sr], Srinivasan obtained the decomposition of the uniform part of W |_{H_{1}×H_{2}} for most of the pairs above. It follows from her results that the Howe duality correspondence respects Lusztig series [A-M-R]. For example, unipotent representations correspond.

Adams and Moy [A-M] then showed that cuspidal unipotent representations correspond in the case of first occurrence in a tower of dual pairs. More recently, Aubert, Michel and Rouquier [A-M-R], starting from the main result in [Sr], were able to describe the correspondence completely, in a combinatorial manner, except in the case of symplectic-orthogonal pairs, for which they stated a conjecture.

The results for the exceptional dual pairs is an analogue for these results, though it is certainly less complete than the classical case. One should perhaps try to obtain an analogue of the result in [Sr].

References

[A-M] J. Adams and A. Moy, Unipotent Representations and Reductive Dual Pairs, Trans.

AMS 340, Pg. 309-321. [A-M-R] M.F. Aubert, J. Michel and R. Rouquier, Correspondance de Howe pour les

Groupes Reductifs sur les Corps Finis, Duke Math. J. 83(1996), Pg. 353-397.

[C] R. Carter, Finite Groups of Lie Type: Conjugacy CLasses and Complex Characters, Wiley(1985).

[D] F. Digne, Descente des Shintani et Restriction des Scalaires, Preprint(1996).

[G] W.T. Gan, Exceptional Dual Pair Correspondence over Finite Fields, preprint(1997).

[Gr] B.H. Gross, The Motive of a Reductive Group, these proceedings. [M-S] K. Magaard and G. Savin, Exceptional Θ-correspondences, to appear in Composi-

tio(1997).

[S] . Srinivasan, Weil Representations of Finite Classical Groups, Invent. Math. 51(1979), Pg. 143-153.