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In this thesis, we explore the representation theory of double affine Hecke algebras (DAHAs) through the lens of stated skein theory. Over the past decade, there have been several works establishing robust connections between skein algebras and DAHAs. Particularly, Samuelson proved that a spherical subalgebra of the type $A_1$ DAHA can be realized as a quotient of the Kauffman bracket skein algebra of the torus with boundary, $K_q(T^2 \setminus D^2)$. Since the $A_1$ double affine Hecke algebra is Morita equivalent to its spherical subalgebra, discovering modules for $K_q(T^2 \setminus D^2)$ immediately provides us with modules for the $A_1$ DAHA.Stated skein theory enhances traditional Kauffman bracket skein theory by incorporating the boundary components of manifolds, thereby offering additional properties such as excision that enrich the algebraic structure. Furthermore, Kauffman bracket skein algebras embed into their stated counterparts, showing that stated skein algebras are extensions of Kauffman bracket skein algebras. We use this extended framework to further develop the representation theory of the $A_1$ DAHA.After identifying generators for the stated skein algebra of $T^2 \setminus D^2$, we embed this algebra into a quantum $6$-torus and leverage the nice representation-theoretic properties of quantum tori to construct a module of Laurent polynomials. Additionally, as $T^2$ is the boundary of any knot complement, we discuss how to construct a more topologically-defined module from various knots and provide an explicit example for the unknot. This approach builds upon the ideas of Berest and Samuelson, who showed that there exists a natural DAHA action on the Kauffman bracket skein module of knot complements.

We construct an isomorphism between the wrapped higher-dimensional Heegaard Floer homology of κ-tuples of cotangent fibers and κ-tuples of conormal bundles of homotopically nontrivial simple closed curves in T ∗Σ with a certain braid skein group, where Σ is a closed oriented surface of genus > 0 and κ is a positive integer. Moreover, we show this produces a (right) module over the surface Hecke algebra associated to Σ. This module structure is shown to be equivalent to the polynomial representation of DAHA in the case where Σ = T 2 and the cotangent fibers and conormal bundles of curves are both parallel copies.

We study the rational Cherednik algebra attached to the complex reflection group $G(r,1,2)$. Each irreducible representation $S^\lambda$ of $G(r,1,2)$ corresponds to a standard module $\Delta(\lambda)$ for the rational Cherednik algebra. We give necessary and sufficient conditions for the existence of morphism between two of these modules and explicit formulas for them when they exist.

In this paper, we define a new presentation for the Dunkl-Opdam subalgebra of the rational Cherednik algebra. This presentation uncovers the Dunkl-Opdam subalgebra as a Drinfeld algebra. We use this fact to define Dirac cohomology for the DO subalgebra. We also formalize generalized graded Hecke algebras and extend a Langlands classification to generalized graded Hecke algebras.

Assume that $\mathbb F$ is an algebraically closed field and let $q$ denote a nonzero scalar in $\mathbb F$ that is not a root of unity. The universal Askey--Wilson algebra $\triangle_q$ is a unital associative $\mathbb F$-algebra defined by generators and relations. The generators are $A,B, C$ and the relations state that each of $$ A+\frac{q BC-q^{-1} CB}{q^2-q^{-2}}, \qquad B+\frac{q CA-q^{-1} AC}{q^2-q^{-2}}, \qquad C+\frac{q AB-q^{-1} BA}{q^2-q^{-2}} $$ is central in $\triangle_q$. The universal DAHA (double affine Hecke algebra) $\mathfrak H_q$ of type $(C_1^\vee,C_1)$ is a unital associative $\mathbb F$-algebra generated by $\{t_i^{\pm 1}\}_{i=0}^3$ and the relations state that \begin{gather*} t_it_i^{-1}=t_i^{-1} t_i=1 \quad \hbox{for all $i=0,1,2,3$}; \\ \hbox{$t_i+t_i^{-1}$ is central} \quad \hbox{for all $i=0,1,2,3$}; \\ t_0t_1t_2t_3=q^{-1}. \end{gather*} Each $\mathfrak H_q$-module is a $\triangle_q$-module by pulling back via the injection $\triangle_q\to \mathfrak H_q$ given by \begin{eqnarray*} A &\mapsto & t_1 t_0+(t_1 t_0)^{-1}, \\ B &\mapsto & t_3 t_0+(t_3 t_0)^{-1}, \\ C &\mapsto & t_2 t_0+(t_2 t_0)^{-1}. \end{eqnarray*} We classify the lattices of $\triangle_q$-submodules of finite-dimensional irreducible $\mathfrak H_q$-modules. As a consequence, for any finite-dimensional irreducible $\mathfrak H_q$-module $V$, the $\triangle_q$-module $V$ is completely reducible if and only if $t_0$ is diagonalizable on $V$., The work gives a q-analog of 1906.09160 and improves 1701.06089

We give a non-negative combinatorial formula, in terms of Littlewood-Richardson numbers, for the homology of the unitary representations of the cyclotomic rational Cherednik algebra, and as a consequence, for the graded Betti numbers for the ideals of a class of subspace arrangements arising from the reflection arrangements of complex reflection groups., 41 pages

In this paper we construct representations of certain graded double affine Hecke algebras (DAHA) with possibly unequal parameters from geometry. More precisely, starting with a simple Lie algebra g \mathfrak {g} together with a Z / m Z \mathbb {Z}/m\mathbb {Z} -grading ⨁ i ∈ Z / m Z g i \bigoplus _{i\in \mathbb {Z}/m\mathbb {Z}}\mathfrak {g}_{i} and a block of D G 0 _ ( g i ) \mathcal {D}_{G_{\underline 0}}(\mathfrak {g}_{i}) as introduced in [J. Represent. Theory 21 (2017), pp. 277-321], we attach a graded DAHA and construct its action on the direct sum of spiral inductions in that block. This generalizes results of Vasserot [Duke Math J. 126 (2005), pp. 251-323] and Oblomkov-Yun [Adv. Math 292 (2016), pp. 601-706] which correspond to the case of the principal block.

For integers $e,\ell\geq 2$, the level $\ell$ Fock space has an $\mathfrak{sl}_\infty$-crystal structure arising from the action of a Heisenberg algebra, intertwining the $\widehat{\mathfrak{sl}_e}$-crystal. The vertices of these crystals are charged $\ell$-partitions. We give the combinatorial rule for computing the arrows anywhere in the $\mathfrak{sl}_\infty$-crystal. This allows us to pinpoint the location of any charged $\ell$-partition. As an application, we compute the support of the spherical representation of a cyclotomic rational Cherednik algebra, and in particular, the set of parameters such that it is finite-dimensional. We also give an easy abacus characterization of all finite-dimensional representations of type $B$ Cherednik algebras., 30 pages, some color figures. New version including the main following changes: rewritten introduction, edited Section 3 (Definitions 3.2, 3.5 and 3.9, proof of Lemma 3.4 and of Theorem 3.14), added references (Remarks 6.6 and 6.15)

We study canonical intertwining operators between modules of the trigonometric Cherednik algebra, induced from the standard modules of the degenerate affine Hecke algebra. We show that these operators correspond to the Zhelobenko operators for the affine Lie algebra $\widehat{\mathfrak{sl}}_m$. To establish the correspondence, we use the functor of Arakawa, Suzuki and Tsuchiya which maps certain $\widehat{\mathfrak{sl}}_m$-modules to modules of the Cherednik algebra., Comment: 26 pages, misprints corrected

With respect to the two-sided cells, we give a decomposition formula for the Kazhdan–Lusztig basis elements of the affine Hecke algebra of type A˜n−1. The main tool is the star operators introduced by Kazhdan and Lusztig. In the appendix, we prove that the left star operator commutes with the right star operator even in the unequal parameter case.

We give a skein-theoretic realization of the $\mathfrak{gl}_n$ double affine Hecke algebra of Cherednik using braids and tangles in the punctured torus. We use this to provide evidence of a relationship we conjecture between the classical skein algebra of the punctured torus and the elliptic Hall algebra of Burban and Schiffmann., Preliminary version, comments welcome! 25 pages, many figures. V2: added some details, diagrams, comments, and subtracted some typos. V3: simplified arguments in Section 3.3, added some clarifying remarks

We consider vertices, a notion originating in local representation theory of finite groups, for the category $\mathcal{O}$ of a rational Cherednik algebra and prove the analogue of the Dipper-Du Conjecture for Hecke algebras of symmetric groups in that setting. As a corollary we obtain a new proof of the Dipper-Du Conjecture over $\mathbb{C}$., 14 pages, last revised March 10, 2021 in accordance with referee report

In this paper we show how the Cherednik algebra of type $\check{C_1}C_1$ appears naturally as quantisation of the group algebra of the monodromy group associated to the sixth Painlevé equation. This fact naturally leads to an embedding of the Cherednik algebra of type $\check{C_1}C_1$ into $Mat(2,\mathbb T_q)$, i.e. $2\times 2$ matrices with entries in the quantum torus. For $q=1$ this result is equivalent to say that the Cherednik algebra of type $\check{C_1}C_1$ is Azumaya of degree 2 [31]. By quantising the action of the braid group and of the Okamoto transformations on the monodromy group associated to the sixth Painlevé equation we study the automorphisms of the Cherednik algebra of type $\check{C_1}C_1$ and conjecture the existence of a new automorphism. Inspired by the confluences of the Painlevé equations, we produce similar embeddings for the confluent Cherednik algebras $\mathcal H_V,\mathcal H_{IV},\mathcal H_{III},\mathcal H_{II}$ and $\mathcal H_{I},$ defined in [27].

We categorify the extended affine Hecke algebra and the affine quantum Schur algebra S(n, r) for 3

We study the polynomial representation of the rational Cherednik algebra of type $A_{n-1}$ with generic parameter in characteristic $p$ for $p \mid n$. We give explicit formulas for generators for the maximal proper graded submodule, show that they cut out a complete intersection, and thus compute the Hilbert series of the irreducible quotient. Our methods are motivated by taking characteristic $p$ analogues of existing characteristic $0$ results., 8 pages. v3: Streamlined proof of complete intersection property in Section 3; main results are unchanged

We classify the simple modules for the rational Cherednik algebra that are irreducible when restricted to W, in the case when W is a finite Weyl group. The classification turns out to be closely related to the cuspidal two-sided cells in the sense of Lusztig. We compute the Dirac cohomology of these modules and use the tools of Dirac theory to find nontrivial relations between the cuspidal Calogero-Moser cells and the cuspidal two-sided cells., Comment: 16 pages; added references, corrected misprints

Let $R$ be a commutative ring, let $F$ be a locally compact non-archimedean field of finite residual field $k$ of characteristic $p$, and let $\mathbf{G}$ be a connected reductive $F$-group. We show that the pro-$p$-Iwahori Hecke $R$-algebra of $G=\mathbf{G}(F)$ admits a presentation similar to the Iwahori–Matsumoto presentation of the Iwahori Hecke algebra of a Chevalley group, and alcove walk bases satisfying Bernstein relations. This was previously known only for a $F$-split group $\mathbf{G}$.

Arakawa, Suzuki and Tsuchiya constructed a correspondence between certain modules of the trigonometric Cherednik algebra $\mathfrak{C}_N$ depending on a parameter $\kappa\in\mathbb{C}$, and certain modules of the affine Lie algebra $\widehat{\mathfrak{sl}}_m$ of level $\kappa-m$. We give a detailed proof of this correspondence by working with the affine Lie algebra $\widehat{\mathfrak{gl}}_m$ alongside of $\widehat{\mathfrak{sl}}_m$. We also relate this construction to a correspondence between certain modules of the degenerate affine Hecke algebra $\mathfrak{H}_N$ and all modules of $\mathfrak{sl}_m$ or $\mathfrak{gl}_m$. The latter correspondence was constructed earlier by Cherednik., Comment: 22 pages

We consider the deformed versions of the classical Howe dual pairs $(O(r),\mathfrak{s}\mathfrak{l}(2))$ and $(O(r),\mathfrak{s}\mathfrak{p}\mathfrak{o}(2|2))$ in the context of a rational Cherednik algebra $H_c=H_c(W,\mathfrak{h})$ associated to a finite Coxeter group $W$ at the parameters $c$ and $t=1$. For the first pair, we compute the centraliser of the well-known copy of $\mathfrak{s}\cong\mathfrak{s}\mathfrak{l}(2)$ inside $H_c$. For the second pair, we show that the classical copy of $\mathfrak{g}\cong\mathfrak{s}\mathfrak{p}\mathfrak{o}(2|2)$ inside the Weyl-Clifford algebra $\mathcal{W}\otimes\mathcal{C}$ deforms to a Lie superalgebra inside $H_c\otimes\mathcal{C}$ and compute its centraliser algebra. For a generic parameter $c$ such that the standard $H_c$-module is unitary, we compute the joint $((H_c)^{\mathfrak{s}},\mathfrak{s})$- and $((H_c\otimes\mathcal{C})^{\mathfrak{g}},\mathfrak{g})$-decompositions of the relevant modules., Comment: 29 pages; version that was accepted for publication. In this revised version we shortened the discussion about Drinfeld orbifold algebras, added a List of symbols and made minor corrections throughout

Properties of relative traces and symmetrizing forms on chains of cyclotomic and affine Hecke algebras are studied. The study relies on the use of bases of these algebras which generalize a normal form for elements of the complex reflection groups G ( m , 1 , n ) , m = 1 , 2 , … , ∞ , constructed by a recursive use of the Coxeter–Todd algorithm. Formulas for inducing, from representations of an algebra in the chain, representations of the next member of the chain are presented.

We study holonomic D -modules on SL n ( C ) × C n , called mirabolic modules , analogous to Lusztig's character sheaves. We describe the supports of simple mirabolic modules. We show that a mirabolic module is killed by the functor of Hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric Cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus. We introduce an analogue of Verdier's specialization functor for representations of Cherednik algebras which agrees, on category O , with the restriction functor of Bezrukavnikov and Etingof. In type A , we also consider a Verdier specialization functor on mirabolic D -modules. We show that Hamiltonian reduction intertwines specialization functors on mirabolic D -modules with the corresponding functors on representations of the Cherednik algebra. This allows us to apply known Hodge-theoretic purity results for nearby cycles in the setting considered by Bezrukavnikov and Etingof.

We introduce parabolic degenerations of rational Cherednik algebras of complex reflection groups, and use them to give necessary conditions for finite-dimensionality of an irreducible lowest weight module for the rational Cherednik algebra of a complex reflection group, and for the existence of a non-zero map between two standard modules. The latter condition reproduces and enhances, in the case of the symmetric group, the combinatorics of cores and dominance order, and in general shows that the c-ordering on category O may be replaced by a much coarser ordering. The former gives a new proof of the classification of finite dimensional irreducible modules for the Cherednik algebra of the symmetric group., 35 pages

In this note, using Nekrasov's gauge origami framework, we study two different versions of the the BPS/CFT correspondence - first, the standard AGT duality and, second, the quiver W algebra construction which has been developed recently by Kimura and Pestun. The gauge origami enables us to work with both dualities simultaneously and find exact matchings between the parameters. In our main example of an A-type quiver gauge theory, we show that the corresponding quiver qW-algebra and its representations are closely related to a large-n limit of spherical gl(n) double affine Hecke algebra whose modules are described by instanton partition functions of a defect quiver theory., Comment: 12 pages, 2 figures, typos corrected

This paper describes irreducible representations in category O of the rational Cherednik algebra H c ( H 3 , h ) associated to the exceptional Coxeter group H 3 and any complex parameter c. We compute the characters of all these representations explicitly. As a consequence, we classify all the finite-dimensional irreducible representations of H c ( H 3 , h ) .

We study equivalences for category Op of the rational Cherednik algebras Hp of type G`(n) = (μ`) n o Sn: a highest weight equivalence between Op and Oσ(p) for σ ∈ S` and an action of S` on a non-empty Zariski open set of parameters p; a derived equivalence between Op and Op′ whenever p and p′ have integral difference; a highest weight equivalence between Op and a parabolic category O for the general linear group, under a non-rationality assumption on the parameter p. As a consequence, we confirm special cases of conjectures of Etingof and of Rouquier.

We study the Dunkl–Opdam subalgebra of the rational Cherednik algebra for wreath products at t = 0 , and use this to describe the block decomposition of restricted rational Cherednik algebras for G ( m , d , n ) .

This paper has two main purposes. Firstly we generalise Ram's explicit construction of calibrated representations of the affine Hecke algebra to the multi-parameter case (including the non-reduced $BC_n$ case). We then derive the Plancherel formulae for all rank~1 and rank~2 affine Hecke algebras including a construction of all representations involved (following work of Opdam)., To appear in Journal of Algebraic Combinatorics

The Calogero-Moser families are partitions of the irreducible characters of a complex reflection group derived from the block structure of the corresponding restricted rational Cherednik algebra. It was conjectured by Martino in 2009 that the generic Calogero-Moser families coincide with the generic Rouquier families, which are derived from the corresponding Hecke algebra. This conjecture is already proven for the whole infinite series G(m,p,n) and for the exceptional group G4. A combination of theoretical facts with explicit computations enables us to determine the generic Calogero-Moser families for the nine exceptional groups G4, G5, G6, G8, G10, G23=H3, G24, G25, and G26. We show that the conjecture holds for all these groups - except surprisingly for the group G25, thus being the first and only-known counter-example so far., Comment: Accepted for publication in Algebras and Representation Theory. In V2: Rewritten introduction and some minor corrections (thanks to the reviewer!). 32 pages. Comments welcome

We introduce families \( \mathcal{B}_n^S\left( {{z_1},\ldots,{z_n}} \right) \) and \( \mathcal{B}_{{n,\hbar}}^S\left( {{z_1},\ldots,{z_n}} \right) \) of maximal commutative subalgebras, called Bethe subalgebras, of the group algebra \( \mathbb{C}\left[ {\mathfrak{S}n} \right] \) of the symmetric group. Bethe subalgebras are deformations of the Gelfand−Zetlin subalgebra of \( \mathbb{C}\left[ {\mathfrak{S}n} \right] \). We describe various properties of Bethe subalgebras.

Let W be a complex reflection group and H_c(W) the Rational Cherednik algebra for $W$ depending on a parameter c. One can consider the category O for H_c(W). We prove a conjecture of Rouquier that the categories O for H_c(W) and H_{c'}(W) are derived equivalent provided the parameters c,c' have integral difference. Two main ingredients of the proof are a connection between the Ringel duality and Harish-Chandra bimodules and an analog of a deformation technique developed by the author and Bezrukavnikov. We also show that some of the derived equivalences we construct are perverse., Comment: 32 pages, preliminary version; v2 33 pages some proofs rewritten; v3 32 pages, proofs of the main results substantially simplified, other minor changes; v4 some proofs were modified, other minor changes

We give an overview of the representation theory of restricted rational Cherednik algebras. These are certain finite-dimensional quotients of rational Cherednik algebras at t=0. Their representation theory is connected to the geometry of the Calogero-Moser space, and there is a lot of evidence that they contain certain information about Hecke algebras even though the precise connection is so far unclear. We outline the basic theory along with some open problems and conjectures, and give explicit results in the cyclic and dihedral cases.

Given a complex reflection group W we compute the support of the spherical irreducible module of the rational Cherednik algebra of W in terms of the simultaneous eigenfunction of the Dunkl operators and Schur elements for finite Hecke algebras., Comment: 21 pages

We give explicit constructions of some finite-dimensional representations of generalized double affine Hecke algebras (GDAHA) of higher rank using $R$-matrices for $U_q(\mathfrak{sl}_N)$. Our construction is motivated by an analogous construction of Silvia Montarani in the rational case. Using the Drinfeld-Kohno theorem for Knizhnik-Zamolodchikov differential equations, we prove that the explicit representations we produce correspond to Montarani's representations under a monodromy functor introduced by Etingof, Gan, and Oblomkov.

We consider ideals of polynomials vanishing on the W-orbits of the intersections of mirrors of a finite reflection group W. We determine all such ideals that are invariant under the action of the corresponding rational Cherednik algebra hence form submodules in the polynomial module. We show that a quantum integrable system can be defined for every such ideal for a real reflection group W. This leads to known and new integrable systems of Calogero–Moser type which we explicitly specify. In the case of classical Coxeter groups, we also obtain generalized Calogero–Moser systems with added quadratic potential.

The theory of congruences for the partition function p ( n ) depends heavily on the properties of half-integral weight Hecke operators. The subject has been complicated by the absence of closed formulas for the Hecke images P ( z ) | T ( l 2 ) , where P ( z ) is the relevant modular generating function. We obtain such formulas using Eulerʼs Pentagonal Number Theorem and the denominator formula for the Monster Lie algebra. As a corollary, we obtain congruences for certain powers of Ramanujanʼs Delta-function.

Let W a be an irreducible affine Weyl group with W 0 the associated Weyl group. The present paper is to study the second lowest two-sided cell Ω qr of W a . Let n qr be the number of left cells of W a in Ω qr . We conjecture that the equality n qr = 1 2 | W 0 | should always hold. When W a is either A ˜ n − 1 , n ⩾ 2 , or of rank ⩽4, this equality can be verified by the existing data (see 0.3). Then the main result of the paper is to prove the inequality n qr ⩽ 1 2 | W 0 | in all cases.

This paper concerns the combinatorics of the orbit Hecke algebra associated with the orbit of a two sided Weyl group action on the Renner monoid of a finite monoid of Lie type, $M$. It is shown by Putcha in \cite{Putcha97} that the Kazhdan-Lusztig involution (\cite{KL79}) can be extended to the orbit Hecke algebra which enables one to define the $R$-polynomials of the intervals contained in a given orbit. Using the $R$-polynomials, we calculate the M\"obius function of the Bruhat-Chevalley ordering on the orbits. Furthermore, we provide a necessary condition for an interval contained in a given orbit to be isomorphic to an interval in some Weyl group., Comment: 12 pages

We apply the Dunkl-Opdam operators and generalized Jack polynomials to study category O for the rational Cherednik algebra of type G(r,1,n). We determine the set of aspherical values, and answer a question of Iain Gordon on the ordering of category O., 20 pages; v2 updated based on reviewer feedback, now with somewhat expanded exposition and proofs

We give a detailed account of a combinatorial construction, due to Cherednik, of cyclic generators for irreducible modules of the affine Hecke algebra of the general linear group with generic parameter q., Latex file, 24 pages, minor corrections. To appear in Journal of Combinatorial Theory A

We introduce the generic central character of an irreducible discrete series representation of an affine Hecke algebra. Using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine Hecke algebras (except for the types E) with arbitrary positive parameters and we prove an explicit product formula for their formal degrees (in all cases)., 68 pages, 5 figures. In the second version an appendix was added

In recent joint work with Wang, we have constructed graded Specht modules for cyclotomic Hecke algebras. In this article, we prove a graded version of the Lascoux-Leclerc-Thibon conjecture, describing the decomposition numbers of graded Specht modules over a field of characteristic zero., Comment: 57 pages; final version