## Unipotent class functions of split special orthogonal groups $$SO^+_{2n}$$ over finite fields.(English)Zbl 0545.20028

Let G be a connected reductive group defined over $$F_ q$$ with a Frobenius morphism F. We can write $$F=jF_ 0$$ where $$F_ 0$$ is a ”split Frobenius” morphism of G and j is an automorphism of finite order.
Thus $$jF_ 0=F_ 0j$$ and there is a maximal torus T of G contained in a Borel subgroup B such that T, B are fixed by F and $$F_ 0$$ and $$F_ 0(t)=t^ q$$ for $$t\in T$$. The groups $$G^{F^ m_ 0}$$, $$G^ F$$ (for an integer $$m\geq 1)$$ of fixed points of G under $$F^ m_ 0$$, F respectively are finite groups on which F, $$F^ m_ 0$$ respectively act as automorphisms. There is a norm map $$n_{F^ m_ 0/F}$$ from the set of F-conjugacy classes of $$G^{F^ m_ 0}$$ (where $$x,y\in G^{F^ m_ 0}$$ are F-conjugate if $$x=g^{-1}yF(g)$$ for some $$g\in G^{F^ m_ 0})$$ onto the set of $$F^ m_ 0$$-conjugacy classes of $$G^ F$$, given by $$a^{-1}F(a)\to F^ m_ 0(a)a^{-1}$$ ($$a\in G)$$. This induces a linear map, also denoted by $$n_{F^ m_ 0/F}$$, from the set of F-class functions on $$G^{F^ m_ 0}$$ (i.e., functions constant on the F-conjugacy classes) onto the set of $$F^ m_ 0$$-class functions on $$G^ F$$. The author takes G to be an even orthogonal group and obtains the striking result that on the space spanned by the unipotent characters, the map $$n_{F_ 0/F}$$ is essentially the ”Fourier transform” defined by G. Lusztig [in Representations of finite Chevalley groups (1978; Zbl 0418.20037)]. In a previous paper [Osaka J. Math. 20, 21-32 (1983; Zbl 0515.20026)] the author studied the zeta functions of the Deligne-Lusztig varieties $$X_ w$$ using the formula $Tr(xF^ m_ 0,\sum_{i}(-1)^ iH^ i_ c(X_ w,\bar Q_{\ell}))=Tr(n_{F/F^ m_ 0}(x)a_ wF,\quad Ind^{G^{F^ m_ 0}}_{B^{F^ m_ 0}}(1)),$ where $$a_ w$$ is a standard basis element of the endomorphism algebra of $$Ind^{G^{F^ m_ 0}}_{B^{F^ m_ 0}}(1)$$. In this paper he uses an analogous formula (1.2.1, 1.3.1) replacing B by a parabolic subgroup, and the varieties $$X_ w$$ by the analogous varieties with respect to a regular subgroup L of G which were defined by G. Lusztig [Invent. Math. 34, 201-213 (1976; Zbl 0371.20039)] and used by him to construct a map $$R^ G_ L$$ taking virtual characters of $$L^ F$$ to virtual characters of $$G^ F$$. The lifting theory of N. Kawanaka [J. Fac. Sci., Univ. Tokyo, Sect. I A 30, 499-516 (1984; Zbl 0536.20023)] is also used, and results of Lusztig on the eigenvalues of $$F_ 0$$ on $$H^ i_ c(X_ w,\bar Q_{\ell})$$ which imply that there is a well-defined $$\lambda_{\rho}=\pm 1$$ attached to every unipotent character $$\rho$$ of $$G^ F$$. Let # be the linear map $$\rho \to \lambda_{\rho}\cdot \rho$$ on the subspace $$C^{(1)}(G^ F)$$ of the space of class functions on $$G^ F$$ spanned by the unipotent characters. The main result of the paper (Theorem 2.8.1) is that $$n_{F_ 0/F}$$ is the Fourier transform modified by #. A corollary (2.8.14) in the case of the $$F_ q$$-split orthogonal group is that $$n_{F_ 0/F_ 0}$$ takes the Fourier transform $${\hat \rho}$$ of a unipotent character to $$\lambda_{\rho}\cdot {\hat \rho}$$. A byproduct of the theorem is the explicit description of the map $$R^ G_ L:C^{(1)}(L^ F)\to C^{(1)}(G^ F)$$ for any regular subgroup L.
Reviewer: B.Srinivasan

### MSC:

 20G05 Representation theory for linear algebraic groups 20G40 Linear algebraic groups over finite fields 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

### Citations:

Zbl 0418.20037; Zbl 0515.20026; Zbl 0371.20039; Zbl 0536.20023
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### References:

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