##
**Finite-dimensional representations of DAHA and affine Springer fibers: the spherical case.**
*(English)*
Zbl 1237.20008

From the introduction: We classify finite-dimensional simple spherical representations of rational double affine Hecke algebras, and we study a remarkable family of finite-dimensional simple spherical representations of double affine Hecke algebras.

Double affine Hecke algebras (DAHA) were introduced by Cherednik about fifteen years ago to prove MacDonald conjectures. The understanding of their representation theory has progressed very much recently, in particular by the classification of the simple modules in the category \(\mathcal O\) by E. Vasserot [in Duke Math. J. 126, No. 2, 251–323 (2005; Zbl 1114.20002)] (when the parameters are not roots of unity). The latter is very similar to Kazhdan-Lusztig classification of simple modules of affine Hecke algebras. One can show that any simple module in the category \(\mathcal O\) is the top of a module induced from an affine Hecke subalgebra (see Corollary A.3.6). However, the representation theory of DAHA has some specific features that have no analogues for affine Hecke algebras (e.g., it is very difficult to classify the finite-dimensional simple modules).

This can be approached in several ways. The DAHA, denoted by \(\mathbf H\), admits two remarkable degenerated forms. The first one, the degenerated DAHA, denoted by \(\mathbf H'\), is an analogue of the degenerate Hecke algebras introduced by Drinfeld and Lusztig. Its representation theory is more or less the same as that of \(\mathbf H\). The second one was introduced by P. Etingof and V. Ginzburg [Invent. Math. 147, No. 2, 243-348 (2002; Zbl 1061.16032)] and is called the rational DAHA (or rational Cherednik algebra). We denote it by \(\mathbf H''\).

In this article, we concentrate on the spherical finite-dimensional modules. The case of nonspherical modules can probably be done with similar techniques. We come back to this issue later. The article contains two main results.

First, we classify all spherical finite-dimensional simple \(\mathbf H''\)-modules in Theorem 2.8.1. Since the finite-dimensional simple \(\mathbf H''\)-modules belong to the category \(\mathcal O\), each of them is the top of a standard module. The spherical ones are the top of a polynomial representation (which is equal to a standard module induced from the trivial representation of the Weyl group). So they are labelled by the value of the parameter of \(\mathbf H''\), which is a rational number \(c=k/m\) with \((k,m)=1\) and \(m>0\). Surprisingly, the classification we get is extremely simple and nice. The spherical finite-dimensional simple modules correspond to the integers \(k,m\) such that \(k<0\) and \(m\) is an elliptic regular number (i.e., the integer \(m\) is the order of an elliptic element of the Weyl group which is regular in Springer’s sense). In type \(E_8\), for instance, there are twelve elliptic regular numbers. The only known cases before were the case where \(m\) is the Coxeter number in arbitrary type and the dihedral types (in particular, all rank 2 types). Notice that in this article we assume that \(\mathbf H''\) is crystallographic with equal parameters. The proof is as follows. Any simple spherical finite-dimensional \(\mathbf H''\)-module \(M''\) also has the structure of a simple spherical \(\mathbf H\)-module, denoted by \(M\). The algebra \(\mathbf H''\) has two remarkable polynomial subalgebras (yielding, under induction, two representations) called the polynomial representations. A spherical finite-dimensional \(\mathbf H''\)-module is a quotient of both polynomial representations. Using this, one can identify \(M\) with the top of a standard \(\mathbf H\)-module with explicit Langlands parameters (see [Vasserot, loc. cit.] for the terminology). Using the Fourier-Sato transform of perverse sheaves shows that this explicit module is finite-dimensional precisely when \(m\) is elliptic regular.

In the second part of the article, we describe explicitly all the spherical Jordan-Hölder factors (modulo a technical hypothesis). This classification (contrarily to the first one) relies on a case-by-case computation. It is quite remarkable that it involves interesting combinatorial objects that already appear in local Langlands correspondence for \(p\)-adic groups. Affine Hecke algebras are related to unramified Langlands correspondence via Bernstein’s functor. DAHAs seem to be related to the tamely ramified correspondence.

The first chapter contains standard facts on elliptic regular elements in Weyl groups, conjugacy classes of tori in \(G\), and homogeneous regular semisimple elements in the loop Lie algebra \(\mathfrak g=\text{Lie}(G_0)\otimes\mathbb C((\varepsilon))\). In particular, Corollary 1.3.3 gives a criterion for the existence of homogeneous elliptic regular semisimple elements in \(\mathfrak g\) which is important for the rest of the article.

In Sections 2.1, 2.2, and 2.3, we recall the definitions and the main properties of DAHAs, degenerate DAHAs, and rational DAHAs. We introduce the category \(\mathcal O\), the polynomial representation, and the spherical modules for each of these algebras. Propositions 2.1.7 and 2.2.4 are analogues of theorems of Lusztig on affine Hecke algebras which compare the categories \(\mathcal O\) of \(\mathbf H\) and \(\mathbf H'\). Proposition 2.3.1 compares the categories \(\mathcal O\) of \(\mathbf H'\) and \(\mathbf H''\).

In Section 2.4, we introduce the affine Springer fibers and the \(\widehat{\mathbf H}\)-action in their homology. The algebra \(\widehat{\mathbf H}\) is another version of \(\mathbf H\) (both algebras are isogenous); it is the one that appears in the geometric picture. The comparison of the modules of \(\mathbf H\) and \(\widehat{\mathbf H}\) is given in Corollary 2.5.8. The simple modules of \(\widehat{\mathbf H}\), \(\mathbf H\) are classified in Proposition 2.5.1 and Theorem 2.5.3, respectively. Section 2.6 contains generalities on the Fourier-Sato transform. In particular, it is related to the Iwahori-Matsumoto involution in Lemma 2.6.1. In Section 2.7, we give a geometric description of the polynomial representation (in Lemma 2.7.2), which yields a simple characterization of the simple finite-dimensional spherical \(\widehat{\mathbf H}\)-modules in Lemma 2.7.3. Section 2.8 contains the main result of the article (i.e., Theorem 2.8.1), which gives a complete list of all simple finite-dimensional spherical \(\mathbf H''\)-modules.

Section 3 contains a description of the simple finite-dimensional spherical modules that appear (with multiplicities) in the homology of affine Springer fibers. First, Theorem 3.3.1 classifies the isomorphism classes of Jordan-Hölder composition factors of the homology of the affine Springer fiber of an elliptic semi-simple regular element of \(\mathfrak g\) in terms of a set \(\mathcal X_{c,1,RS}\) of local systems. Then Theorem 3.3.6 yields an explicit description of the set \(\mathcal X_{c,1,RS}\) under the technical hypothesis of Conjecture 3.3.3. The proof of Theorem 3.3.6 is based on an explicit description of the affine Springer map in the case we are interested in. It uses the technical results in Sections 3.1 and 3.2, which are proved via the evaluation map \(\mathbb C((\varepsilon))\to\mathbb C\), \(f(\varepsilon)\mapsto f(1)\). The assumption of Conjecture 3.3.3 is checked in a large number of cases in Section 4 (on a case-by-case analysis).

Double affine Hecke algebras (DAHA) were introduced by Cherednik about fifteen years ago to prove MacDonald conjectures. The understanding of their representation theory has progressed very much recently, in particular by the classification of the simple modules in the category \(\mathcal O\) by E. Vasserot [in Duke Math. J. 126, No. 2, 251–323 (2005; Zbl 1114.20002)] (when the parameters are not roots of unity). The latter is very similar to Kazhdan-Lusztig classification of simple modules of affine Hecke algebras. One can show that any simple module in the category \(\mathcal O\) is the top of a module induced from an affine Hecke subalgebra (see Corollary A.3.6). However, the representation theory of DAHA has some specific features that have no analogues for affine Hecke algebras (e.g., it is very difficult to classify the finite-dimensional simple modules).

This can be approached in several ways. The DAHA, denoted by \(\mathbf H\), admits two remarkable degenerated forms. The first one, the degenerated DAHA, denoted by \(\mathbf H'\), is an analogue of the degenerate Hecke algebras introduced by Drinfeld and Lusztig. Its representation theory is more or less the same as that of \(\mathbf H\). The second one was introduced by P. Etingof and V. Ginzburg [Invent. Math. 147, No. 2, 243-348 (2002; Zbl 1061.16032)] and is called the rational DAHA (or rational Cherednik algebra). We denote it by \(\mathbf H''\).

In this article, we concentrate on the spherical finite-dimensional modules. The case of nonspherical modules can probably be done with similar techniques. We come back to this issue later. The article contains two main results.

First, we classify all spherical finite-dimensional simple \(\mathbf H''\)-modules in Theorem 2.8.1. Since the finite-dimensional simple \(\mathbf H''\)-modules belong to the category \(\mathcal O\), each of them is the top of a standard module. The spherical ones are the top of a polynomial representation (which is equal to a standard module induced from the trivial representation of the Weyl group). So they are labelled by the value of the parameter of \(\mathbf H''\), which is a rational number \(c=k/m\) with \((k,m)=1\) and \(m>0\). Surprisingly, the classification we get is extremely simple and nice. The spherical finite-dimensional simple modules correspond to the integers \(k,m\) such that \(k<0\) and \(m\) is an elliptic regular number (i.e., the integer \(m\) is the order of an elliptic element of the Weyl group which is regular in Springer’s sense). In type \(E_8\), for instance, there are twelve elliptic regular numbers. The only known cases before were the case where \(m\) is the Coxeter number in arbitrary type and the dihedral types (in particular, all rank 2 types). Notice that in this article we assume that \(\mathbf H''\) is crystallographic with equal parameters. The proof is as follows. Any simple spherical finite-dimensional \(\mathbf H''\)-module \(M''\) also has the structure of a simple spherical \(\mathbf H\)-module, denoted by \(M\). The algebra \(\mathbf H''\) has two remarkable polynomial subalgebras (yielding, under induction, two representations) called the polynomial representations. A spherical finite-dimensional \(\mathbf H''\)-module is a quotient of both polynomial representations. Using this, one can identify \(M\) with the top of a standard \(\mathbf H\)-module with explicit Langlands parameters (see [Vasserot, loc. cit.] for the terminology). Using the Fourier-Sato transform of perverse sheaves shows that this explicit module is finite-dimensional precisely when \(m\) is elliptic regular.

In the second part of the article, we describe explicitly all the spherical Jordan-Hölder factors (modulo a technical hypothesis). This classification (contrarily to the first one) relies on a case-by-case computation. It is quite remarkable that it involves interesting combinatorial objects that already appear in local Langlands correspondence for \(p\)-adic groups. Affine Hecke algebras are related to unramified Langlands correspondence via Bernstein’s functor. DAHAs seem to be related to the tamely ramified correspondence.

The first chapter contains standard facts on elliptic regular elements in Weyl groups, conjugacy classes of tori in \(G\), and homogeneous regular semisimple elements in the loop Lie algebra \(\mathfrak g=\text{Lie}(G_0)\otimes\mathbb C((\varepsilon))\). In particular, Corollary 1.3.3 gives a criterion for the existence of homogeneous elliptic regular semisimple elements in \(\mathfrak g\) which is important for the rest of the article.

In Sections 2.1, 2.2, and 2.3, we recall the definitions and the main properties of DAHAs, degenerate DAHAs, and rational DAHAs. We introduce the category \(\mathcal O\), the polynomial representation, and the spherical modules for each of these algebras. Propositions 2.1.7 and 2.2.4 are analogues of theorems of Lusztig on affine Hecke algebras which compare the categories \(\mathcal O\) of \(\mathbf H\) and \(\mathbf H'\). Proposition 2.3.1 compares the categories \(\mathcal O\) of \(\mathbf H'\) and \(\mathbf H''\).

In Section 2.4, we introduce the affine Springer fibers and the \(\widehat{\mathbf H}\)-action in their homology. The algebra \(\widehat{\mathbf H}\) is another version of \(\mathbf H\) (both algebras are isogenous); it is the one that appears in the geometric picture. The comparison of the modules of \(\mathbf H\) and \(\widehat{\mathbf H}\) is given in Corollary 2.5.8. The simple modules of \(\widehat{\mathbf H}\), \(\mathbf H\) are classified in Proposition 2.5.1 and Theorem 2.5.3, respectively. Section 2.6 contains generalities on the Fourier-Sato transform. In particular, it is related to the Iwahori-Matsumoto involution in Lemma 2.6.1. In Section 2.7, we give a geometric description of the polynomial representation (in Lemma 2.7.2), which yields a simple characterization of the simple finite-dimensional spherical \(\widehat{\mathbf H}\)-modules in Lemma 2.7.3. Section 2.8 contains the main result of the article (i.e., Theorem 2.8.1), which gives a complete list of all simple finite-dimensional spherical \(\mathbf H''\)-modules.

Section 3 contains a description of the simple finite-dimensional spherical modules that appear (with multiplicities) in the homology of affine Springer fibers. First, Theorem 3.3.1 classifies the isomorphism classes of Jordan-Hölder composition factors of the homology of the affine Springer fiber of an elliptic semi-simple regular element of \(\mathfrak g\) in terms of a set \(\mathcal X_{c,1,RS}\) of local systems. Then Theorem 3.3.6 yields an explicit description of the set \(\mathcal X_{c,1,RS}\) under the technical hypothesis of Conjecture 3.3.3. The proof of Theorem 3.3.6 is based on an explicit description of the affine Springer map in the case we are interested in. It uses the technical results in Sections 3.1 and 3.2, which are proved via the evaluation map \(\mathbb C((\varepsilon))\to\mathbb C\), \(f(\varepsilon)\mapsto f(1)\). The assumption of Conjecture 3.3.3 is checked in a large number of cases in Section 4 (on a case-by-case analysis).

### MSC:

20C08 | Hecke algebras and their representations |

14M15 | Grassmannians, Schubert varieties, flag manifolds |

### Keywords:

double affine Hecke algebras; degenerate Hecke algebras; simple modules in category \(\mathcal O\); simple spherical representations; induced modules; rational Cherednik algebras; spherical finite-dimensional modules
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\textit{M. Varagnolo} and \textit{E. Vasserot}, Duke Math. J. 147, No. 3, 439--540 (2009; Zbl 1237.20008)

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