Proof of the Deligne-Langlands conjecture for Hecke algebras. (English) Zbl 0613.22004

In this paper, the authors are concerned with a special case of the general Langlands conjecture on irreducible representations of reductive p-adic groups.
Let \(G\) be a connected reductive algebraic group over \({\mathbb{C}}\) with simply connected derived group; u be a unipotent element of \(G\); \((s,q)\) be a semi-simple element of \(G\times {\mathbb{C}}^{\times}\) such that \(sus^{-1}=u^q\). Let \(M(u)=\{(g,q)\in G\times {\mathbb{C}}^{\times}\); \(gug^{-1}=u^q\}^a \)closed subgroup of \(G\times {\mathbb{C}}^{\times}\); \(M(u,s)=\{(g,a)\in M(u)\); \(g\in Z(s)\}\) an algebraic subgroup of \(M(u)\); \(\bar M(u,s)=M(u,s)/M(u,s)^0\), where \(M(u,s)^0\) is the connected component of \(M(u,s)\) containing the identity. Let \(M\) be the smallest algebraic subgroup of \(G\times {\mathbb{C}}^{\times}\) containing \((s,q)\), \(R_M\) be the Grothendieck group of \(M\) and \({\mathbb{R}}_M=R_M\otimes {\mathbb{C}}\). Let \({\mathbb{H}}\) be the abstract algebra over \({\mathbb{C}}[q,q^{- 1}]\) called Hecke algebra which has a natural homomorphism into the Iwahori-Matsumoto algebra of the extended affine Weyl group.
For an irreducible complex representation \(\rho\) of \(\bar M(u,s)\), the authors construct the “standard \({\mathbb{H}}\)-module” \({\mathcal M}_{u,s,q,\rho}=\rho^*\otimes_{\mathbb{C}}({\mathbb{C}}_{s,q}\otimes_{\mathbb{R}_M}K^M_0({\mathcal B}_u))^{\bar M(u,s)}\), where \({\mathbb{C}}_{s,q}={\mathbb{C}}\) is a natural \({\mathbb{R}}_M\)-module corresponding to (s,q) and \(K^M_0({\mathcal B}_u)\) is the K-homology on the variety \({\mathcal B}_u\) of all Borel subgroups containing u. The procedure is dual to that of the authors’ previous work [ibid. 80, 209–231 (1985; see the preceding review)] in the sense that here they use equivariant K-homology instead of equivariant K-cohomology. Further, they prove that if q is not a root of unity and \(\rho\) appears in the natural representation of \(\bar M(u,s)\) on \(K_0({\mathcal B}^s_u)\), where \({\mathcal B}^s_u=\{B\in {\mathcal B}_u\), \(s\in B\}\), then the standard \({\mathbb{H}}\)-module \({\mathcal M}_{u,s,q,\rho}\) has a unique simple quotient which depends only on the G-conjugacy class of \((u,s,\rho)\); conversely, any simple \({\mathbb{H}}\)-module in which q acts as multiplication by q is obtained from some \((u,s,\rho)\) as above, and \((u,s,\rho)\) is uniquely determined by the given \({\mathbb{H}}\)-module up to G-conjugacy. An important role in the proof is played by the analysis of the equivariant K-homology of the space of triples \((u,B,B')\) with \(B,B'\in {\mathcal B}_u\).
The authors also give the classification of tempered and square integrable representations of \({\mathbb{H}}\).
Reviewer: E.Abe


22E50 Representations of Lie and linear algebraic groups over local fields
11F33 Congruences for modular and \(p\)-adic modular forms
14G20 Local ground fields in algebraic geometry


Zbl 0613.22003
Full Text: DOI EuDML


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