1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is equal to the number of characters χ ∈ Irr(Sn) such that χAn is not irreducible. (Hint. Consider the vector space of class functions on An which are invariant under conjugation by the transposition (12).)

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Character Theory of Finite Groups

An introduction to noncommutative noetherian rings

Introduction Hilbert scheme of points Framed moduli space of torsion free sheaves on $\mathbb{P}^2$ Hyper-Kahler metric on $(\mathbb{C}^2)^{[n]}$ Resolution of simple singularities Poincare polynomials of the Hilbert schemes (1) Poincare polynomials of Hilbert schemes (2) Hilbert scheme on the cotangent bundle of a Riemann surface Homology group of the Hilbert schemes and the Heisenberg algebra Symmetric products of an embedded curve, symmetric functions and vertex operators Bibliography Index.

Lectures on Hilbert schemes of points on surfaces

https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/107169/1/0183-7.pdf

Equivariant Completion (代数幾何学の研究)

We re-examine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic resolution While some modification from this classical context is necessary, many familiar features survive These include a version of the Beilinson-Bernstein localization theorem, a theory of Harish-Chandra bimodules and their relationship to convolution operators on cohomology, and a discrete group action on the derived category of representations, generalizing the braid group action on category O via twisting functors 
Our primary goal is to apply these results to other quantized symplectic resolutions, including quiver varieties and hypertoric varieties This provides a new context for known results about Lie algebras, Cherednik algebras, finite W-algebras, and hypertoric enveloping algebras, while also pointing to the study of new algebras arising from more general resolutions

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Quantizations of conical symplectic resolutions I: local and global structure

These are the lecture notes for the 5M reading course ”Lie groups, Lie algebras, and their representations” at the University of Glasgow, autumn 2015.

https://www.maths.gla.ac.uk/~gbellamy/lie.pdf

Lie groups, Lie algebras, and their representations

Using combinatorial properties of complex reflection groups, we show that the generalised Calogero-Moser space associated to the centre of the corresponding rational Cherednik algebra is singular for all values of its deformation parameter c if and only if the group is different from the wreath product $S_n\wr C_m$ and the binary tetrahedral group. This result and a theorem of Ginzburg and Kaledin imply that there does not exist a symplectic resolution of the singular symplectic variety h+h*/W outside of these cases; conversely we show that there exists a symplectic resolution for the binary tetrahedral group (Hilbert schemes provide resolutions for the wreath product case).

https://www.maths.gla.ac.uk/~gbellamy/fakes/singular.pdf

On singular Calogero-Moser spaces

Using combinatorial properties of complex reflection groups, we show that the generalised Calogero-Moser space associated to the centre of the corresponding rational Cherednik algebra is singular for all values of its deformation parameter c if and only if the group is different from the wreath product $S_n\wr C_m$ and the binary tetrahedral group This result and a theorem of Ginzburg and Kaledin imply that there does not exist a symplectic resolution of the singular symplectic variety h+h*/W outside of these cases; conversely we show that there exists a symplectic resolution for the binary tetrahedral group (Hilbert schemes provide resolutions for the wreath product case)

In this article, we consider Nakajima quiver varieties from the point of view of symplectic algebraic geometry. We prove that they are all symplectic singularities in the sense of Beauville and completely classify which admit symplectic resolutions. Moreover we show that the smooth locus coincides with the locus of canonically $\theta$-polystable points, generalizing a result of Le Bruyn; we study their etale local structure and find their symplectic leaves. An interesting consequence of our results is that not all symplectic resolutions of quiver varieties appear to come from variation of GIT.

Symplectic resolutions of quiver varieties

We show that the quotient C4/G admits a symplectic resolution for ${G = Q_8 \times_{{\bf Z}/2} D_8 < {\sf Sp}_4({\bf C})}$. Here Q8 is the quaternionic group of order eight and D8 is the dihedral group of order eight, and G is the quotient of their direct product which identifies the nontrivial central elements −Id of each. It is equipped with the tensor product representation ${{\bf C}^2 \boxtimes {\bf C}^2 \cong {\bf C}^4}$. This group is also naturally a subgroup of the wreath product group ${Q_8^2 \rtimes S_2 < {\sf Sp}_4({\bf C})}$. We compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming C4/G. We also discuss preliminary investigations on the more general question of classifying linear quotients V/G admitting symplectic resolutions.

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Gwyn Bellamy

Papers

Lie groups, Lie algebras, and their representations

On singular Calogero-Moser spaces

On singular Calogero-Moser spaces

Symplectic resolutions of quiver varieties

A new linear quotient of C 4 admitting a symplectic resolution