On the classification of irreducible representations of affine Hecke algebras with unequal parameters. (Q2889344)

Given a based root datum \(\mathcal R=(X,R_0,Y,R_0^\vee,F_0)\) with finite Weyl group \(W_0\) (generated by the reflections \(s_\alpha\) for \(\alpha\in R_0\)) and extended affine Weyl group \(W^e=X\rtimes W_0\), and a parameter function \(q\colon W^e\to\mathbb C^\times\), there is an \textit{affine Hecke algebra} \(\mathcal H=\mathcal H(\mathcal R,q)\). Let \(R_{\text{nr}}=R_0\cup\{2\alpha:\alpha^\vee\in 2Y\}\).NEWLINENEWLINE One speaks of \textit{positive parameters} (resp. of \textit{equal parameters}) if \(q\) takes positive real values (resp. the same value) on all simple (affine) reflections in \(W^e\). In the equal parameters case, if the value of \(q\) is a power of a prime number \(p\), then \(\mathcal H\) is isomorphic to the convolution algebra of the Iwahori-biinvariant functions on a reductive \(p\)-adic group with root datum \(\mathcal R^\vee=(Y,R_0^\vee,X,R_0,F_0^\vee)\). As shown by Lusztig, affine Hecke algebras with unequal parameters also arise as intertwining algebras in the smooth representation theory of reductive \(p\)-adic groups.NEWLINENEWLINE Let \(\Gamma\) be a finite group of diagram automorphisms of \((R_0,F_0)\) and assume that \(q_{\alpha^\vee}=q_{\gamma(\alpha^\vee)}\) for all \(\gamma\in\Gamma\) and \(\alpha\in R_{\text{nr}}\), where \(\alpha^\vee\mapsto q_{\alpha^\vee}\) denotes the \(W_0\)-invariant map \(R_{\text{nr}}^\vee\to\mathbb C^\times\) which is naturally induced by \(q\). Then \(\Gamma\) acts on \(\mathcal H\) by algebra automorphisms and one can form the crossed product algebra \(\mathcal H\rtimes\Gamma\), whose natural basis is indexed by the group \((X\rtimes W_0)\rtimes\Gamma=X\rtimes(W_0\rtimes\Gamma)\). We call the algebra \(\mathcal H\rtimes\Gamma\) an \textit{extended affine Hecke algebra}. Extended affine Hecke algebras are an important tool for the classification of both the smooth dual and the unitary smooth dual of a reductive \(p\)-adic group.NEWLINENEWLINE The article under review provides a classification of the dual of the algebra \(\mathcal H\rtimes\Gamma\). In such a generality (that is, possibly unequal parameters without any condition, except the positivity ones) this result is completely new. (The reason for the assumption that the parameters are positive is due to the approach used by the author which is based on harmonic analysis on extended affine Hecke algebras.)NEWLINENEWLINE The algebra \(\mathcal H\) is a deformation of the group algebra \(\mathbb C[W^e]\), so it is natural to compare the representation theory of \(\mathcal H\) and that of \(W^e\), and, more generally, to compare the representation theory of \(\mathcal H\rtimes\Gamma\) and that of \(W^e\rtimes\Gamma\).NEWLINENEWLINE The author defines a map \(\pi\mapsto\text{Spr}(\pi)\) from irreducible \(\mathcal H\rtimes\Gamma\)-representations to \(W^e\rtimes\Gamma\)-representations, which can be viewed as a kind of ``affine Springer correspondence''. Although this map does not preserve irreducibility it has a lot of nice properties, in particular, the collection of representations \(\text{Spr}(\pi)\) forms a \(\mathbb Q\)-basis of the representation ring of \(W^e\rtimes\Gamma\) (Theorem~2.3.1 in the paper).NEWLINENEWLINE Then the author adjusts the construction of the above map in order to get a continuous bijection NEWLINE\[NEWLINE\text{Irr}(W^e\rtimes\Gamma)\to\text{Irr}(\mathcal H\rtimes\Gamma)NEWLINE\]NEWLINE from the dual space of \(W^e\rtimes\Gamma\) to that of \(\mathcal H\rtimes\Gamma\), as we will now explain.NEWLINENEWLINE First, a model for \(\text{Irr}(W^e\rtimes\Gamma)\) is provided by the \textit{extended quotient} of \(T\) by \(W'=W_0\rtimes\Gamma\), where \(T\) is the complex torus \(\Hom_{\mathbb Z}(X,\mathbb C^\times)\). Consider the \(W'\)-action on \(W'\times T\) defined by \(w'\cdot(w,t)=(w'ww^{\prime-1},w't)\) and define NEWLINE\[NEWLINE\widetilde T=\{(w,t)\in W'\times T:w\cdot t=t\}.NEWLINE\]NEWLINE Then the extended quotient of \(T\) by \(W'\) is defined as the (ordinary) quotient \(\widetilde T/W'\). Let \(\mathcal Q(\mathcal R)\) denote the variety of all maps \(v\colon R_{\text{nr}}^\vee/W'\to\mathbb C^\times\). To every \(v\in\mathcal Q(\mathcal R)\) one associates the parameter function \(q_{\alpha^\vee}=v(\alpha^\vee)^2\).NEWLINENEWLINE Then it is proved in the paper (Theorem~5.4.2) that there exists a continuous bijection NEWLINE\[NEWLINE\mu\colon\widetilde T/W'\to\text{Irr}({\mathcal H}\rtimes\Gamma)NEWLINE\]NEWLINE and a map \(h\colon T/W'\times\mathcal Q(\mathcal R)\to T\) such that the following holds:NEWLINENEWLINE \(\bullet\) \(h\) is locally constant in the first argument;NEWLINENEWLINE \(\bullet\) for fixed \(c\in\widetilde T/W'\), \(h(c,v)\) is a monomial in the variables \(v(s)^{\pm 1}\), where \(s\) runs through all simple affine reflections;NEWLINENEWLINE \(\bullet\) the central character of \(\mu(W'(w,t))\) is \(W'h(W'(w,t),q^{1/2})t\).NEWLINENEWLINE The above result proves the validity in a large number of cases of a conjecture of Baum, Plymen and the reviewer for representations of reductive \(p\)-adic groups.NEWLINENEWLINE The proofs of Theorems 2.3.1 and 5.4.2 rely on several difficult and technical intermediate results which have independent interest, too. Among these results, there are:NEWLINENEWLINE (1) A \textit{Langlands classification} for irreducible representations of extended affine Hecke algebras, which reduces the problem of the determination of the smooth dual to those of the \textit{tempered duals of parabolic subalgebras} of \(\mathcal H(\mathcal R,q)\rtimes\Gamma\) (Theorem~2.2.4).NEWLINENEWLINE (2) A very interesting connection of the representation theory of \(\mathcal H(\mathcal R,q)\rtimes\Gamma\) with that of \(\mathcal H(\mathcal R,q^\varepsilon)\rtimes\Gamma\) with \(\varepsilon\in\mathbb R\) (Corollary~4.2.2), and of the representation theory of the Schwartz algebra \(\mathcal S(\mathcal R,q)\rtimes\Gamma\) with that of \(\mathcal S(\mathcal R,q^\varepsilon)\rtimes\Gamma\) (Theorem~4.4.2).NEWLINENEWLINE (3) As a special case of (2), the construction of an injective homomorphism NEWLINE\[NEWLINE\zeta_0\colon\mathcal S(W^e\rtimes\Gamma)=\mathcal S(\mathcal R,q^0)\rtimes\Gamma\to\mathcal S(\mathcal R,q)\rtimes\Gamma,NEWLINE\]NEWLINE which has the property that NEWLINE\[NEWLINE\pi\circ\zeta_0\simeq\text{Spr}(\pi)NEWLINE\]NEWLINE for every irreducible tempered representation \(\pi\) of \(\mathcal H(\mathcal R,q)\rtimes\Gamma\).NEWLINENEWLINE (4) The demonstration that the Schwartz algebras \(\mathcal S(\mathcal R,q)\rtimes\Gamma\) and \(\mathcal S(W^e\rtimes\Gamma)\) are \textit{geometrically equivalent}.NEWLINENEWLINE (5) The proof of a conjecture of Higson and Plymen, which says that the \(K\)-theory of the \(C^*\)-completion of an affine Hecke algebra \(\mathcal H(\mathcal R,q)\) does not depend on the parameter function \(q\).NEWLINENEWLINE Moreover, in the last section, the author provides a very good illustration of his main results in the case of affine Hecke algebra with \(R_0\) of type \(B_2/C_2\) and \(X\) the root lattice.