These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of the book by N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry , Birkhäuser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal enveloping algebra of a complex semisimple Lie algebra, a quantum group or the ...

A solution to the problem of representing compositional structure using distributed representations is described. The method uses circular convolution to associate items, which are represented by vectors. Arbitrary variable bindings, short sequences of various lengths, frames, and reduced representations can be compressed into a xed width vector. These representations are items in their own rig...

For locally compact groups Gi, i = 1, 2, · · · , n, let CB(G1, · · · , Gn) denote the Banach space of completely bounded multilinear forms on C0(G1)×· · ·×C0(Gn) in the completely bounded norm. CB(G1, · · · , Gn) has the structure of a Banach ∗-algebra under a multiplication and adjoint operation which agree with the convolution structure on the measure algebra M(G1 ×· · ·×Gn). If the Gi are al...

Let K be a compact Lie group acting by automorphisms on a nilpotent Lie group N . One calls (K,N) a Gelfand pair when the integrable K-invariant functions on N form a commutative algebra under convolution. We prove that in this case the coadjoint orbits for G := K nN which meet the annihilator k⊥ of the Lie algebra k of K do so in single K-orbits. This generalizes a result of the authors and R....

A b s t r a c t. Let Wp,q be the Fourier modulation space FMp,q and let ∗σ be the twisted convolution. If a ∈ D′ such that (a ∗σ φ,φ) ≥ 0 for every φ ∈ C∞ 0 , and χ ∈ S such that χ(0) 6= 0, then we prove that χa ∈ Wp,∞ iff a ∈ Wp,∞. We also present some extensions to the case when weighted Fourier modulation spaces are used. AMS Mathematics Subject Classification (2000): 47B65, 35A21, 35S05

Following Braverman–Finkelberg–Feigin–Rybnikov (arXiv: 1008.3655), we study the convolution algebra of a handsaw quiver variety, a.k.a. a parabolic Laumon space, and a finite W -algebra of type A. This is a finite analog of the AGT conjecture on 4-dimensional supersymmetric Yang–Mills theory with surface operators. Our new observation is that the C-fixed point set of a handsaw quiver variety is...

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of the book N. Chriss, V. Ginzburg ‘Representation Theory and Complex Geometry’ , Birkhäuser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal enveloping algebra of a complex semisimple Lie algebra, a Quantum group or the Iwaho...

We introduce the categories of infinitesimal Hopf modules and bimodules over an infinitesimal bialgebra. We show that they correspond to modules and bimodules over the infinitesimal version of the double. We show that there is a natural, but non-obvious way to construct a pre-Lie algebra from an arbitrary infinitesimal bialgebra and a dendriform algebra from a quasitriangular infinitesimal bial...

In this paper we investigate a ring of delay-differential operators, which appears in the study of time-delay systems with commensurate point-delays. Besides point-delay differential operators, this ring contains also certain distributed delays. It can be described as a certain algebraic extension of the ring of point-delay differential operators as well as a convolution algebra of compact supp...