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The Hecke algebra of a reductive \(p\)-adic group: a geometric conjecture. (English) Zbl 1120.14001
Consani, Caterina (ed.) et al., Noncommutative geometry and number theory. Where arithmetic meets geometry and physics. Based on two workshops, Bonn, Germany, August 2003 and June 2004. Wiesbaden: Vieweg (ISBN 3-8348-0170-4/hbk). Aspects of Mathematics E 37, 1-34 (2006).
The authors formulate a conjecture for the ideals in the Bernstein decomposition of the Hecke algebra \[ \mathcal H(G) := \bigcup_K\mathcal H(G//K) \] of a reductive \(p\)-adic group \(G\). Here, \(K\) ranges among the compact open subgroups of \(G\), and \(\mathcal H(G//K)\) is the convolution algebra of all complex-valued, compactly supported functions on \(G\) such that \(f(k_1xk_2) = f(x)\) for all \(k_1,k_2\in K\).
\(\mathcal H(G)\) is a noncommutative, non-unital, non finitely-generated, non reduced \(\mathbb C\)-algebra and so cannot be the coordinate ring of an affine algebraic variety.
The Bernstein decomposition \[ \mathcal H(G) = \bigoplus_{\mathfrak s\in\mathfrak B(G)}\mathcal H^{\mathfrak s}(G) \] [cf. J. Bernstein, P. Deligne, D. Kazhdan and M.-F. Vignéras, “Représentations des groupes réductifs sur un corps local” (1984; Zbl 0544.00007)] is a canonical decomposition of the Hecke algebra into two-sided ideals \(\mathcal H^{\mathfrak s}(G)\). \(\mathfrak B(G)\) denotes the set of equivalence classes \([(L,\sigma)]_G\) of cuspidal pairs \((L,\sigma)\), where \(L\) is a Levi subgroup of \(G\) and \(\sigma\) is an irreducible, supercuspidal representation of \(L\). Two pairs are said to be equivalent \((L,\sigma)\sim (L',\sigma')\), if \(gLg^{-1} = L'\) and \({}^g\sigma \sim \nu'\sigma'\), for some \(g\in G\) and some \(\nu'\in \Psi(L')\), where \(\Psi(L)\) is the group of unramified quasicharacters of \(L\).
Although each ideal \(\mathcal H^{\mathfrak s}(G)\) is a non-unital, non finitely-generated, non reduced \(\mathbb C\)-algebra, it admits a set of local units. These units are constructed from the set of local units of the Hecke algebra. The local units for \(\mathcal H(G)\) are idempotent elements \(e_K\) of the algebra such that for any given finite subset \(X\subset \mathcal H(G)\), \(e_Kx = x = xe_K\) for all \(x\in X\).
Using this fact, the authors prove their first result (Theorem 1) claiming that each ideal \(\mathcal H^{\mathfrak s}(G)\) is a \(k\)-algebra Morita equivalent to a unital \(k\)-algebra of finite type, where \(k\) is the coordinate ring of a complex affine algebraic variety (a complex torus).
Thus, one may think of an ideal \(\mathcal H^{\mathfrak s}(G)\) as a noncommutative algebraic variety and \(\mathcal H(G)\) as a noncommutative scheme.
The conjecture states that each ideal \(\mathcal H^{\mathfrak s}(G)\) is geometrically equivalent to the coordinate ring of a complex affine algebraic variety \(X^{\mathfrak s}\): \(\mathcal H^{\mathfrak s}(G) \asymp \mathcal O(X^{\mathfrak s})\).
This conjecture is closely related to Lusztig’s conjecture on the asymptotic Hecke algebra and its canonical decomposition [G. Lusztig, Hecke algebras with unequal parameters. CRM Monograph Series 18, Am. Math. Soc. (2003; Zbl 1051.20003)].
The equivalence relation of geometric equivalence \(\asymp\) is defined on the collection of associative \(\mathbb C\)-algebras \(A\) which are countably generated \(k\)-algebras, i.e. which admit a finite or countable basis as \(\mathbb C\)-vector spaces. Here, \(k = \mathcal O(X)\) denotes the coordinate ring of a complex affine algebraic variety.
Given two such \(k\)-algebras \(A,B\): \(A\asymp B\) if (at least) one of the following three cases holds:
(1) [Morita equivalence of \(k\)-algebras with local units]. Assume that \(A\) and \(B\) have a countable set of local units. Then, \(A\asymp B\) if \(A\) and \(B\) are connected by a strict Morita context [cf. N. Chifan, S. Dascalescu and C. Nastasescu, J. Algebra 284, No. 2, 705–736 (2005; Zbl 1111.16007)].
(2) [Spectrum preserving morphisms with respect to filtrations of \(k\)-algebras of finite type]. Assume that \(A\) and \(B\) are \(k\)-algebras of finite type. Then, \(A\asymp B\) if there exists a morphism \(\phi: A \to B\) of \(k\)-algebras which is either spectrum preserving (i.e. \(\phi\) determines a bijection between the corresponding sets of primitive ideals) or more in general, spectrum preserving with respect to (finite) filtrations on the algebras, in a suitable sense [cf. P. Baum and V. Nistor, K-Theory 27, No. 4, 329–357 (2002; Zbl 1056.16005)].
(3) [Deformation of central character]. Assume that \(A\) is a unital algebra over \(\mathbb C\) which comes endowed with a composed map of \(k\)-algebras \(k \to A[t^{\pm 1}] \to A\), where the second arrow is the evaluation of a Laurent polynomial \(P(t)\) at a non-zero complex number \(q\): \(P(t) \mapsto P(q)\). Denote by \(A(q)\) the finite type \(k\)-algebra so obtained. Then, \(A(q_1)\asymp A(q_2)\), if \(q_1, q_2\) are any two non-zero complex numbers.
\((1), (2), (3)\) generate the notion of geometric equivalence. It is known [cf. Baum and Nistor (loc. cit.); J. Cuntz, K-Theory 15, No. 4, 301–305 (1998; Zbl 0920.18007)] that \[ A\asymp B \quad\Longrightarrow\quad HP_*(A) \cong HP_*(B), \] that is geometric equivalent algebras have isomorphic periodic cyclic homology groups.
The authors succeed to prove the conjecture in several cases.
First of all, the conjecture is proven for all generic points \(\mathfrak s\in \mathfrak B(G)\) in the Bernstein spectrum (Theorem 2).
Examples of generic points are \(\mathfrak s = [G,\sigma]_G\), with \(\sigma\) an irreducible supercuspidal representation of \(G\), and \(\mathfrak s = [\text{GL}(2)\times \text{GL}(2), \sigma_1\otimes\sigma_2]_{\text{GL}(4)}\), with \(\sigma_1\nsim\sigma_2\) (after unramified twist).
Then, the conjecture is proven when \(G = \text{SL}(2,F)\) (Theorem 3), \(F=\) non-archimedean local field, by working out separately the cases of \(\mathfrak s\) non-generic (i.e., \(\mathfrak s = \mathfrak i\) Iwahori ideal, and \(\mathfrak s = \mathfrak j = [T,\lambda]_{\text{SL}(2)}\), \(T\subset \text{SL}(2)\) maximal torus, \(\lambda: \left(\begin{smallmatrix} x & 0\\0 & x^{-1}\end{smallmatrix}\right)\mapsto \chi(x)\), \(\chi=\) unitary character of \(F^\times\) of order two), and then applying Theorem 2 to the generic points.
The conjecture is also verified when \(G = \text{GL}(n,F)\) (Theorem 5), and for the Iwahori ideals of the groups \(\text{PGL}(n)\) (Theorem 6) and \(\text{SO}(5)\) (Theorem 7).
For the entire collection see [Zbl 1089.58001].

MSC:
14A22 Noncommutative algebraic geometry
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16D90 Module categories in associative algebras
20C15 Ordinary representations and characters
20C08 Hecke algebras and their representations
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