##
**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.

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 |

### Keywords:

connected reductive group; Frobenius morphism; maximal torus; Borel subgroup; automorphisms; norm map; class functions; even orthogonal group; unipotent characters; zeta functions; Deligne-Lusztig varieties; endomorphism algebra; parabolic subgroup; regular subgroup; virtual characters; Fourier transform
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### References:

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