Green functions and a conjecture of Geck and Malle
Let \(W\) be a finite Coxeter group. \textit{G.~Lusztig} has attached to each irreducible representation \(E\) of \(W\) a polynomial \(D_E\in\mathbb{R}[X]\) (where \(X\) is an indeterminate) called the formal or generic degree of \(E\) and two functions: the \(a\)-function is defined as \(E\mapsto a_E\), where \(a_E\) is the greatest integer \(n\geq 0\) such that \(X^n\) divides the polynomial \(D_E\), the \(b\)-function \(b(E)\) is defined as \(E\mapsto b_E\) where \(b_E\) is the smallest integer \(m\geq 0\) such that \(E\) occurs with non-zero multiplicity in \(V^{\otimes m}\), where \(V\) is an \(\mathbb{R} W\)-module affording the reflection character of \(W\) (see [Characters of reductive groups over a finite field, Ann. Math. Stud. 107 (1984; Zbl 0556.20033)]). We always have \(a_E\leq b_E\), and \(E\) is called special if the equality holds. The set \(\text{Irr}(W)\) of isomorphism classes of irreducible representations of \(W\) is partioned into various two-sided cells of \(W\). Each two-sided cell contains a unique special representation of \(W\). Let \(G\) be a connected reductive group defined over an algebraic closure \(k\) of a finite field \(\mathbb{F}_q\) with \(q\) elements. Assume that \(W\) is the Weyl group of \(G\). The Springer correspondence defines a natural injective map from \(\text{Irr}(W)\) to the set of the pairs \((C,{\mathcal L})\), where \(C\) is a unipotent class in \(G\) and \(\mathcal L\) is a \(G\)-equivariant irreducible \(\overline\mathbb{Q}_\ell\)-local system on \(C\). A unipotent class \(C\) is called special if \((C,\overline\mathbb{Q}_\ell)\) corresponds to a special representation of \(W\). For any special unipotent class \(C\) in \(G\), the \(\mathcal L\) special piece \(\widehat C\) is defined as the subset of the unipotent variety \(G_{\text{uni}}\) consisting of all the elements in the closure \(\overline C\) of \(C\) which are not in the closure of any special unipotent class \(C'\) such that \(C'\subset\overline C\) and \(C'\neq C\). Hence \(\widehat C\) is an irreducible, locally closed subvariety in \(G_{\text{uni}}\). It is known that special pieces form a partition of \(G_{\text{uni}}\). We assume that \(G\) has a split \(\mathbb{F}_q\)-structure with Frobenius endomorphism \(F\). Then all the unipotent classes are \(F\)-stable. It is known that, once the characteristic \(p\) of \(\mathbb{F}_q\) is fixed, the cardinality of the \(F\)-fixed points in a unipotent class \(C\) can be expressed as a polynomial in \(q\). The classification of unipotent classes depends on \(p\). Netherveless, \textit{G.~Lusztig} has shown [in Asian J. Math. 1, No. 1, 194-207 (1997; Zbl 0929.20034)] that if \(C\) is the (special) unipotent class corresponding to a special representation \(E\) of \(W\), and \(\widehat C\) is the special piece associated to \(C\), then there exists a polynomial \(f_E\in\mathbb{Z}[X]\) such that \(|\widehat C^F|=f_E(q)\), that is, the polynomial expressing the cardinality \(|\widehat C^F|\) of \(\widehat C^F\) does not depend on \(p\). In [Exp. Math. 8, No. 3, 281-290 (1999; Zbl 0962.20033)] \textit{M.~Geck} and \textit{G.~Malle} proposed a general algorithm for producing polynomials \(f_E(q)\) without referring to unipotent classes. Their algorithm is an analog of \textit{G.~Lusztig}'s algorithm for computing the Green functions (see [Adv. Math. 61, 103-155 (1986; Zbl 0602.20036)]), and is given as follows. Let \(S_W\) be the coinvariant algebra of the symmetric algebra \(S(V)\). Then \(S_W=\bigoplus_{m\geq 0}S_W^m\) is a graded \(\mathbb{R} W\)-module. We define, for any representation \(E\) of \(W\), a polynomial \(R(E)\in\mathbb{Z}[X]\) by \[ R(E):=\sum_{m\geq 0}\dim\Hom_W(E,S_W^m) X^m. \] For \(i\) in the image \({\mathcal I}_0\) of the Springer correspondence, we denote by \(E_i\) the corresponding irreducible representation of \(W\). We put \(a(i):=a_{E_i}\) and we define a matrix \(\Omega=(\omega_{ij})_{i,j\in{\mathcal I}_0}\) by \[ \omega_{ij}:=X^N R(E_i\otimes E_j\otimes\varepsilon)\in\mathbb{Z}[X], \] where \(N\) is the number of reflections in \(W\), and \(\varepsilon\) is the sign character of \(W\). We consider the following system of equations for unknown \(\lambda_{ij}\), \(p_{ij}\) (\(i,j\in{\mathcal I}_0\)): \[ \begin{cases}\lambda_{i'j'}=0&\text{unless \(a(i')=a(j')\)};\\ p_{j'j}=0&\text{unless \(a(j')>a(j)\) or \(j=j'\)};\\ p_{jj}=X^{a(j)}&\text{for all \(j\in{\mathcal I}_0\)};\\ \omega_{ij}=\sum_{i',j'\in{\mathcal I}_0}p_{i'i}\lambda_{i'j'}p_{j'j}&\text{for all \(i,j\in{\mathcal I}_0\)}.\end{cases}\tag{*} \] The system of equations (*) has a unique solution with \(\lambda_{ij}\), \(p_{ij}\) in \(\mathbb{Q}(X)\). Under this setting M.~Geck and G.~Malle stated the following conjecture (and verified it for all exceptional groups): \textbf{Conjecture 1:} (Geck, Malle) (a) \(\lambda_{ij}\), \(p_{ij}\in\mathbb{Q}[X]\) for \(i,j\in{\mathcal I}_0\); (b) \(\sum_i\lambda_{ii}=X^{2N}\), where the sum is taken over all \(i\in{\mathcal I}_0\) such that \(E_i\) is special; (c) for \(i\in{\mathcal I}_0\) such that \(E_i\) is special, \(\lambda_{ii}\) coincides with \(f_{E_i}(X)\). In the case of type \(A_n\), the equations (*) are exactly the same as the equations for computing the Green functions for \(G\), and the conjecture follows from the properties of the Green functions. For a unipotent class \(C\) in \(G\), choose \(u\in C\) and let \(A_G(u)\) be the component group of the centralizer of \(u\) in \(G\). Then the \(G\)-equivariant irreducible local systems on \(C\) are in bijective correspondence with the isomorphism classes of irreducible representations of \(A_G(u)\). Let \({\mathcal B}_u\) be the variety of Borel subgroups of \(G\) containing \(u\). We have an action of \(W\times A_G(u)\) on the \(\ell\)-adic cohomology group \(H^m({\mathcal B}_u,\overline\mathbb{Q}_\ell)\) (the Springer action of \(W\) and the natural action of \(A_G(u)\)). Assume that \((C,{\mathcal L})\) is represented by \((u,\rho)\). Then the \(\rho\)-isotypic subspace \(H^m({\mathcal B}_u,\overline\mathbb{Q}_\ell)_\rho\) of \(H^m({\mathcal B}_u,\overline\mathbb{Q}_\ell)\) is a \(W\)-module. It is known that the pair \((C,{\mathcal L})\) is in \({\mathcal I}_0\) if and only if \(H^{2d_u}({\mathcal B}_u,\overline\mathbb{Q}_\ell)_\rho\neq 0\), where \(d_u\) is the dimension of the variety \({\mathcal B}_u\), and in this case the \(W\times A_G(u)\)-module \(H^{2d_u}({\mathcal B}_u,\overline\mathbb{Q}_\ell)_\rho\) can be written as \(E\otimes\rho\) for any irreducible representation \(E\) of \(W\). The correspondence \((C,{\mathcal L})\mapsto E\) gives rise to a bijection between \({\mathcal I}_0\) and \(\text{Irr}(W)\) which is nothing but the Springer correspondence. Note that the condition \((C,{\mathcal L})\in{\mathcal I}_0\) is also equivalent to the condition \(H^m({\mathcal B}_u,\overline\mathbb{Q}_\ell)_\rho\neq 0\) for some \(m\geq 0\) by \textit{G.~Lusztig} [Adv. Math. (loc. cit.), 24.4]. Concerning the relationship between Springer representations and the \(a\)-function the followings are known. \(\bullet\) Assume that \((u,1)\) corresponds to \(E\). Then \(d_u=b_E\geq a_E\) (see \textit{W.~Borho, R.~MacPherson} [C. R. Acad. Sci., Paris, Sér. I 292, 707-710 (1981; Zbl 0467.20036)]). In particular, if \(E\) is special, then we have \(d_u=a_E\). \(\bullet\) More generally, if \((u,\rho)\) corresponds to \(E\), then we have \(d_u\geq a_E\) (see \textit{M.~Kashiwara, T.~Tanisaki} [Invent. Math. 77, 185-198 (1984; Zbl 0611.22008)] and \textit{G.~Lusztig} [J. Algebra 104, 146-194 (1986; Zbl 0603.20037)]). In the paper under review, T.~Shoji states the following conjecture, which gives a connection between the \(a\)-function and the Springer representations occuring in lower cohomology: \textbf{Conjecture 2:} (Shoji) Assume that \(E\) corresponds to \((C,{\mathcal L})\) and that \(E_1\) corresponds to \((C_1,{\mathcal L}_1)=(u_1,\rho_1)\) with \(C_1\subset\overline C\). (a) If \(E\) occurs in \(H^m({\mathcal B}_{u_1},\overline\mathbb{Q}_\ell)_{\rho_1}\) for some \(m\), then \(a_{E_1}\geq a_E\). (b) In the setting of (a), assume further that \(C\) is special with \({\mathcal L}=\overline\mathbb{Q}_\ell\) and that \(C_1\not\subset\widehat C\). Then \(a_{E_1}>a_E\). T.~Shoji proves the conjecture~2 for \(G\) of type \(B_n\), \(C_n\) and \(D_n\) with \(n\leq 6\), and for \(G\) of exceptional type, assuming that \(p\) is good for \(G\). (The first assertion in the case of exceptional groups is proved as a consequence of the following stronger result: Let \(G\) be of exceptional type, and assume that \(p\) is good for \(G\). Then, if \(E\), \(E_1\) correspond respectively to \((C,{\mathcal L})\), \((C_1,{\mathcal L}_1)\) with \(C_1\subset\overline C\) and \(C_1\neq\overline C\), then we have \(a_{E_1}\geq a_E\). Note that this result does not hold for classical groups in general, counterexamples are given in the paper). The main result of the paper is the following: Let \(G\) be of classical type, and assume that \(p\) is odd. Assume further that the conjecture~2 holds for \(G\). Then \(\lambda_{ij}\), \(p_{ij}\in\mathbb{Q}[X,X^{-1}]\). Moreover, the statements~(b), (c) in the conjecture~1 hold for \(\lambda_{ii}\). The idea of the proof is to compare the equations in (*) with the equations used to compute the Green functions, which are obtained by expressing the Green functions in terms of certain functions, associated to pairs \((C,{\mathcal L})\in{\mathcal I}_0\), on the set of unipotent elements. These functions form a basis of the space \({\mathcal V}_0\) of all the functions which are generated by the Green functions. T.~Shoji constructs a new basis of the space \({\mathcal V}_0\) which is closely related with the \(a\)-function. Then the result is obtained by expressing the Green functions in terms of this new basis.
scientific article; zbMATH DE number 1428666