## 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

### MathOverflow Questions:

A question on the representations of affine Hecke algebras

### MSC:

 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
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### References:

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