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Exceptional Howe correspondences over finite fields. (English) Zbl 0939.20010
Let $$k$$ be a finite field with $$q$$ elements and of odd characteristic $$p$$. Recall that if $$\text{Sp}_{2N}(q)$$ denotes a symplectic group over $$k$$, the Weil representation of $$\text{Sp}_{2N}(q)$$ defines a correspondence (called the Howe correspondence) between the irreducible representations of two subgroups $$G$$ and $$G'$$ of $$\text{Sp}_{2N}(q)$$ which are mutual commutants in it. Examples of such pairs $$(G,G')$$ are $$(\text{Sp}_{2n},O_m)$$, $$(\text{GL}_n,\text{GL}_m)$$, $$(U_n,U_m)$$. B. Srinivasan [in Invent. Math. 51, 143–153 (1979; Zbl 0399.20040)] determined the decomposition of the uniform part of the Weil representation into Deligne-Lusztig characters of $$G\cdot G'$$. From her results follows the fact that the Howe correspondence respects the correspondence between the Lusztig series induced by some natural inclusion of the dual groups $$\iota\colon G^*\hookrightarrow G^{\prime*}$$ or $$\iota'\colon G^{\prime*}\hookrightarrow G^*$$. In particular, the unipotent representations correspond. J. Adams and A. Moy proved [in Trans. Am. Math. Soc. 340, No. 1, 309–321 (1993; Zbl 0829.22025)] that the cuspidal unipotent representations correspond in the case of first occurrence. The Howe correspondence over finite fields was also studied by the reviewer and J. Michel and R. Rouquier [in Duke Math. J. 83, No. 2, 353–397 (1996; Zbl 0856.22027)].
In the paper under review, the author studies an analogue of the Howe correspondence over finite fields which involves exceptional groups instead of classical ones. Assuming that $$\mathbf H$$ is a split simply-laced linear algebraic group defined over $$k$$, the author first considers an analogue of the Weil representation. This is the so-called reflection representation $$\Pi$$, which has been studied by G. Lusztig [in Lond. Math. Soc. Lect. Note Ser. 34, 325–337 (1979; Zbl 0426.20034)], and which is a unipotent principal series representation of the group $$H$$ of $$k$$-points of $$\mathbf H$$ and whose dimension is the smallest among the dimensions of all nontrivial irreducible representations of $$H$$. It is the analogue over finite fields of the minimal representation studied by D. Kazhdan and G. Savin [in Isr. Math. Conf. Proc. 2, 209–223 (1990; Zbl 0737.22008)].
In the paper under review, it is shown that the character values of the representation $$\Pi$$ on a semisimple element $$\sigma$$ of $$H$$ can be interpreted in terms of the motive $$M(\sigma)$$, introduced by B. Gross, of the connected component of the centralizer of $$\sigma$$ in $$H$$. More precisely, if $$F$$ denotes the Frobenius map, and $$\sigma$$ is $$F$$-stable, we have $$\text{Tr}(\sigma|\Pi)=\text{Tr}(F|M(\sigma))$$.
The author considers various dual pairs in exceptional groups. Among them, there is the pair $$(\text{PGL}_3,G_2)$$ in $$H$$, where $$H$$ is a split adjoint group of type $$E_6$$. This pair is interesting because there is a natural inclusion between the dual groups $$\iota\colon\text{SL}_2\hookrightarrow G_2$$. For each irreducible representation $$\pi$$ of the group $$\text{PGL}_3(q)$$, let $$\Theta(\pi)$$ denote the set of irreducible representations $$\pi'$$ of the group $$G_2$$, counted with multiplicities, such that $$\pi\otimes\pi'$$ occurs in the restriction of $$\text{PGL}_3(q)\cdot G_2$$ of the representation $$\Pi$$ of $$H$$. Also let $$\Theta_{\text{gen}}(\pi)$$ be the subset of $$\Theta(\pi)$$ of generic representations. Then the set $$\Theta_{\text{gen}}(\pi)$$ is determined for each generic representation $$\pi$$ of $$\text{PGL}_3(q)$$. Recall that the isomorphism classes of generic representations of a connected reductive group with connected center can be parametrized by semisimple classes in the dual group. For a semisimple class $$(s)$$ in the dual group, let $$\pi_{(s)}$$ denote the corresponding generic representation. Then the author proves that $$\Theta_{\text{gen}}(\pi_{(s)})=\{\pi_{(\iota(s))}\}$$ for each semisimple element $$s$$ in $$\text{SL}_3(q)$$, the dual group of $$\text{PGL}_3(q)$$. He also has similar results for the dual pairs $$(G_2,\text{PGSp}_6)$$ in $$E_7$$ and $$(G_2,\text{PU}_3)$$ in $$^2E_6$$. These results are in a natural way complements to those of G. Savin and the author [in Can. J. Math. 51, No. 1, 130–146 (1999; Zbl 0933.22014)], of B. Gross and G. Savin [in Compos. Math. 114, No. 2, 153–217 (1998; Zbl 0931.11015)] and of K. Magaard and G. Savin [in Compos. Math. 107, No. 1, 89–123 (1997; Zbl 0878.22011)], where the correspondence of tempered spherical representations was determined for the $$p$$-adic dual pair $$(G_2,\text{PGSp}_6)$$ in the split of type $$E_7$$.

##### MSC:
 20C33 Representations of finite groups of Lie type 20G40 Linear algebraic groups over finite fields 20G05 Representation theory for linear algebraic groups
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