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The subject of this thesis is the interplay between the geometry and the representation theory of rational Cherednik algebras at t = 0. Exploiting this relationship, we use representation theoretic techniques to classify all complex re ection groups for which the geometric space associated to a rational Cherednik algebra, the generalized Calogero-Moser space, is singular. Applying results of Ginzburg-Kaledin and Namikawa, this classification allows us to deduce a (nearly complete) classification of those symplectic reflection groups for which there exist crepant resolutions of the corresponding symplectic quotient singularity. Then we explore a particular way of relating the representation theory and geometry of a rational Cherednik algebra associated to a group W to the representation theory and geometry of a rational Cherednik algebra associated to a parabolic subgroup of W. The key result that makes this construction possible is a recent result of Bezrukavnikov and Etingof on completions of rational Cherednik algebras. This leads to the definition of cuspidal representations and we show that it is possible to reduce the problem of studying all the simple modules of the rational Cherednik algebra to the study of these nitely many cuspidal modules. We also look at how the Etingof-Ginzburg sheaf on the generalized Calogero-Moser space can be "factored" in terms of parabolic subgroups when it is restricted to particular subvarieties. In particular, we are able to confirm a conjecture of Etingof and Ginzburg on "factorizations" of the Etingof-Ginzburg sheaf. Finally, we use Clifford theoretic techniques to show that it is possible to deduce the Calogero-Moser partition of the irreducible representations of the complex reflection groups G(m; d; n) from the corresponding partition for G(m; 1; n). This confirms, in the case W = G(m; d; n), a conjecture of Gordon and Martino relating the Calogero-Moser partition to Rouquier families for the corresponding cyclotomic Hecke algebra.

This thesis presents the results of an investigation into the representation theory of the alternating and dihedral groups and explores how their irreducible representations can be distinguished with the use of class sums.

This thesis is divided into two parts. In the first, we generalize results of Greenberg-Vatsal on the behavior of algebraic lambda-invariants of p-ordinary modular forms under congruence. In the second, we generalize a result of Emerton on maps between locally algebraic parabolically induced representations and unitary Banach space representations of GL₂ over a p-adic field. / text

This thesis is a study in Representation Theory and Geometry. These two branches of mathematics have a fruitful interaction, with many applications to Physics and other sciences. Central objects of interest are homogeneous spaces and their symmetry groups. The geometric and analytic properties of homogeneous spaces relate to the structure and representation theory of the corresponding groups. The focus of this work is on certain representation theoretic phenomena related to equivariant embeddings of homogeneous spaces. The Borel-Weil-Bott theorem, a milestone in Representation Theory and Geometry, provides realizations for every irreducible module of a semisimple complex Lie group as various cohomology spaces of homogeneous vector bundles on flag manifolds of the group. The purpose of this dissertation is to initiate a study of the behaviour of the Borel-Weil-Bott construction under pullbacks along equivariant embeddings of flag manifolds. These pullbacks provide certain geometric branching rules for representations. This is where the notion of a cohomological component arises from. The central result of the dissertation is a criterion for nonvanishing of the pullback. The framework used for the formulation and proof of the result is Kostant's theory of Lie algebra cohomology. After the general criterion is established, various specializations and applications are presented: special classes of embeddings are considered, for which the criterion takes simpler forms; relations are established between pullbacks along embeddings of complete and partial flag manifolds; properties of the set of cohomological components are obtained; various examples are considered, the most interesting of which is related to the theory of invariants of semisimple Lie algebras. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2011-08-23 11:55:57.845

The purpose of this paper is to introduce the reader to Lie algebras and representation theory.

We study representations of Khovanov-Lauda-Rouquier (KLR) algebras of affine Lie type. Associated to every convex preorder on the set of positive roots is a system of cuspidal modules for the KLR algebra. For a balanced order, we study imaginary semicuspidal modules by means of `imaginary Schur-Weyl duality'. We then generalize this theory from balanced to arbitrary convex preorders for affine ADE types. Under the assumption that the characteristic of the ground field is greater than some explicit bound, we prove that KLR algebras are properly stratified. We introduce affine zigzag algebras and prove that these are Morita equivalent to arbitrary imaginary strata if the characteristic of the ground field is greater than the bound mentioned above. Finally, working in finite or affine affine type A, we show that skew Specht modules may be defined over the KLR algebra, and real cuspidal modules associated to a balanced convex preorder are skew Specht modules for certain explicit hook shapes.

For each finite dimensional Lie algebra $\mathfrak{g}$ and positive integer $k$ there exists a modular tensor category $\mathcal{C}(\mathfrak{g},k)$ consisting of highest weight integrable $\hat{\mathfrak{g}}$-modules of level $k$ where $\hat{\mathfrak{g}}$ is the corresponding affine Lie algebra. Relations between the classes $[\mathcal{C}(\mathfrak{sl}_2,k)]$ in the Witt group of nondegenerate braided fusion categories have been completely described in the work of Davydov, Nikshych, and Ostrik. Here we give a complete classification of relations between the classes $[\mathcal{C}(\mathfrak{sl}_3,k)]$ relying on the classification of conncted \'etale alegbras in $\mathcal(\mathfrak_3,k)$ ($SU(3)$ modular invariants) given by Gannon. We then give an upper bound on the levels for which exceptional connected \'etale algebras may exist in the remaining rank 2 cases ($\mathcal{C}(\mathfrak{so}_5,k)$ and $\mathcal{C}(\mathfrak{g}_2,k)$) in hopes of a future classification of Witt group relations among the classes $[\mathcal{C}(\mathfrak{so}_5,k)]$ and $[\mathcal{C}(\mathfrak{g}_2,k)]$. This dissertation contains previously published material.