# Parabolic restriction and induction for rational DAHA’s.

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## Rational double affine Hecke algerbas.

The rational double affine Hecke algebras (=rational DAHA’s), also called rational Cherednik algebras, have been introduced by Etingof and Ginzburg [EG02]. Let us recall their definition. Definition 2.1.1. Let c be a map from S to C that is constant on the W-conjugacy classes. The rational double affine Hecke algebra Hc(W, h) is the quotient of the smash product of CW and the tensor algebra of h h by the relations [x, x0] = 0, [y, y0] = 0, [y, x] = hx, yi − X s2S cshs, yihx, _ s is, for all x, x0 2 h, y, y0 2 h. Here h·, ·i is the canonical pairing between h and h, the element s is a generator of Im(s|h − 1) and _ s is the generator of Im(s|h − 1) such that hs, _ s i = 2.

### Hom-space for functors on Oc(W, h).

Let Projc(W, h) denote the full subcategory of Oc(W, h) consisting of projective objects.
Let I : Projc(W, h) ! Oc(W, h) denote the canonical embedding functor. The following lemma will be useful to us. Lemma 2.3.1. For any abelian category A and any right exact functors F1, F2 from Oc(W, h) to A, the homomorphism of vector spaces rI : Hom(F1, F2) ! Hom(F1 I, F2 I), 7! 1I.

#### Parabolic restriction and induction for rational DAHA’s.

From now on we assume that hW = 1. Fix a point b 2 h. Let W0 be the parabolic subgroup of W given by the stabilizer of b. We will use the same notation as in Section 1.2. Let c0 : S0 ! C be the map given by the restriction of c : S ! C, and consider the rational DAHA Hc0(W0, h) and the category Oc0(W0, h). In [BE09], Bezrukavnikov and Etingof defined the parabolic restriction and induction functors Resb : Oc(W, h) ! Oc0(W0, h) , Indb : Oc0(W0, h) ! Oc(W, h). In this section, we will first review their constructions (Sections 3.1–3.3), then we give some further properties of these functors (Sections 3.4–3.8).

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The isomorphism and the equivalence R.

For a point p 2 h we write C[[h]]p for the completion of C[h] at p, and we write dC[h]p for the completion of C[h] at the W-orbit of p in h. Note that we have C[[h]]0 = dC[h]0. For any C[h]-module M we write cMp = dC[h]p C[h] M. The completions bHc(W, h)b, bHc0(W0, h)0 are well defined algebras. We denote by bOc(W, h)b the category of bHc(W, h)b-modules that are finitely generated over dC[h]b, and we denote by bOc0(W0, h)0 the category of bHc0(W0, h)0-modules that are finitely generated over dC[h]0. Let P = FunW0(W, bHc(W0, h)0) be the set of W0-invariant maps from W to bHc(W0, h)0. Let Z(W,W0, bHc(W0, h)0) be the ring of endomorphisms of the right bHc(W0, h)0-module P. The following is due to Bezrukavnikov and Etingof [BE09, Theorem 3.2].

Introduction
1 Contexte
2 Présentation des résultats
I Crystals and rational DAHA’s
1 Hecke algebras
2 Category O of rational DAHA’s
3 Parabolic restriction and induction for rational DAHA’s.
4 Fock spaces and cyclotomic rational DAHA’s
5 Categorifications and crystals
II Canonical bases and affine Hecke algebras of type D.
2 Affine Hecke algebras of type D
3 Global bases of V and projective graded R-modules
IIIThe v-Schur algebras and Jantzen filtration
1 Statement of the main result
2 Jantzen filtration of standard modules
3 Affine parabolic category O and v-Schur algebras
4 Generalities on D-modules on ind-schemes
5 Localization theorem for affine Lie algebras of negative level
6 The geometric construction of the Jantzen filtration
7 Proof of the main theorem
A List of Notation
Bibliography

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