Exceptional Howe correspondences over finite fields.

*(English)*Zbl 0939.20010Let \(k\) 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 \(E_7\).

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 \(E_7\).

Reviewer: Anne-Marie Aubert (Paris)

##### MSC:

20C33 | Representations of finite groups of Lie type |

20G40 | Linear algebraic groups over finite fields |

20G05 | Representation theory for linear algebraic groups |