
Loop groups, Langlands and mathematical physics

Semester I 2015 
Lecturer: Arun Ram, 174 Richard Berry, email: aram@unimelb.edu.au
Time and Location:
Wednesdays 10:0011:30 In Room 107 Richard Berry
Description
 (a) The lectures will be independent of each other, assuming that that audience has not seen any of the other lectures.
 (b) The lectures will (at least partially) address a number of requests/suggestions I have had from various students and colleagues.
 (c) The lectures will help me to put together a number of things I have been studying
in my effort to tie together Macdonald polynomials, representations of affine Lie algebras,
and elliptic cohomology of flag varieties.
Plan of the lectures

25 February 2015
Title: Geometric Langlands: The fundamental diagram
handwritten lecture notes (pdf file).
Abstract: In this talk we present the diagram of the Satake and Whittaker correspondences
as a structural underpinning of the Geometric Langlands program. I will follow
the general framework as presented in my paper with Kendra Nelsen.

4 March 2015
Title: Central extensions: Heisenberg, Virasoro and affine Lie algebras
handwritten lecture notes (pdf file).
Abstract: This talk reviews the definitions of central extensions,
and universal central extensions, give examples of Heisenberg groups,
groups of order p^{3}, Virasoro algebras, affine Lie algebras,
and then give a presentation of the metaplectic groups using what is sometimes
called the "tame symbol". This talk is an effort to process Chapter 7 of Steinberg's Yale Lecture Notes.

11 March 2015
Title: Twisted Loop groups
handwritten lecture notes (pdf file),
accompanying tables (pdf file).
Abstract: This talk will review the construction and generators of twisted Chevalley groups, including
the source of nonreduced root systems, the root subgroups of twsted Chevalley groups and
the Bruhat decomposition. The general idea is to present some effort to process chapter 11
of Steinberg's Yale Lecture notes. If time permits we will also treat the affine case following Kac's book
on Infinite dimensional Lie algebras and BruhatTits classification of reductive groups over local fields.

18 March 2015
Title: Conformal blocks
handwritten lecture notes (pdf file).
Abstract: In this talk we review the construction of conformal blocks from the representations
of affine Lie algebras, and the path to the KnizhnikZamolodchikov connection by way of
the Virasoro algebra action coming from Sugawara operators. This talk is an effort to process
Chapter 4 of the book of ShimizuUeno "Topics in moduli theory".

25 March 2015
Title: Virasoro representations
handwritten lecture notes (pdf file).
Abstract: A conformal field theory is tightly connected to Virasoro representations (almost equivalent) and the GoddardKentOlive construction (also called the coset construction) is a good way to construct some Virasoro representations. This talk will be a survey of the representation theory of Virasoro algebras, with a description of the coset construction. The general idea is to process the last part of Chapter 12 of Kac's book on Infinite dimensional Lie algebras and the last chapter of Wakimoto's book on infinite dimensional Lie algebras (somewhat from the point of view of FeiginFuchs). A good reference is Chapter 3 of the PhD thesis of Emilie Wiesner.

1 April 2015
Title: Solitons
handwritten lecture notes (pdf file).
Abstract: This will be a survey of the connection between affine Lie algebras, solitons,
and the KP and KdV hierarchies of differential equations. Playing a fundamental role in this story are the irreducible representation of highest weight
Λ_{0}, the orbit of the highest weight vector, the tau functions,
and the Hirota derivatives. The point is that the orbit of the highest
weight vector in L(Λ_{0}) is essentially the loop Grassmannian
and the KP hierachy is a way of expressing the "Plücker relations" of the
loop Grassmannian as a subvariety of the projective space P(L(Λ_{0})).
The general idea for this lecture is to process Chapter 14 of Kac's book on
Infinite dimensional Lie algebras (with a lot of help from Miwa, Jimbo and Date).

8 April 2015
Title: Affine Lie algebras and modular forms
handwritten lecture notes (pdf file).
Abstract:
This will be a survey of the connection between the characters and
modular forms. I will explain the indexing of highest weight representations,
of irreducible integrable representation and how the characters of the
integrable representations are written in terms of theta functions.
This provides a theta function version of the fundamental diagram for
Geometric Langlands. If time permits I'll give the SL(2,Z) action on the
characters of level m integrable representations. The general idea is to
process Chapter 13 of Kac's book on Infinite dimensional Lie algebras.

15 April 2015
Title: Fusion rings
handwritten lecture notes (pdf file).
Abstract:
This talk will be a survey of the connection between Verlinde algebras, representations of loop groups and fusion, and the combinatorics of symmetric functions. I will explain what the coefficients are, their relation to ordinary Weyl characters, and how they are categorfied by
conformal blocks. The general idea is to process exercises 13.34 and 13.35 in Kac, section 4.5 of ShimizuUeno, Chapter 5 of Wakimoto’s book,
and a 2009 paper of Boysal and Kumar.

22 April 2015
Title: Walgebras
handwritten lecture notes (pdf file).
Abstract:
I will define Slodowy slices, finite and affine Walgebras, and explain some conversions between. Following
Arakawa’s recent paper arXiv:1211.7124 we will construct the BRST/quantised DrinfeldSokolov reduction for each case,
Poisson algebras, associative algebras, and vertex algebras, in such a way that the analogy between cases is clear.
The Walgebras are endomorphism algebras of “GelfanGraev” or “Whittaker” modules, and are generalisations of the
Virasoro algebra.
and a 2009 paper of Boysal and Kumar.
 Title: Quantum cohomology of flag varieties
Abstract: This talk will be a survey of the MIT lecture notes of D. Peterson, describing the isomorphism
between the Schubert calculus of loop groups and the quantum cohomology of flag varieties. I also
hope to explain a bit some relation to this and the recent work of Korff (and coauthors) relating the quantum cohomology of flag varieties and the fusion ring (Verlinde algebra).
 Title: Walgebras
Abstract: This talk will be a survey of the point of view on Walgebras taken in the book of
E. Frenkel, “Langlands correspondence for Loop groups”. I will endeavour to explain the relationship
between these algebras, symmetric functions, the HarishChandra isomorphism, and Whittaker modules.
 Title: Springer fibers
Abstract: In this talk I will explain GoreskyKottwitzMacpherson’s beautiful method of labeling
the points of affine Springer fibers by constructing them as towers of affine spaces sitting above a
Hessenberg variety. I will also outline how Hikita has used this to make a connection to Macdonald polynomials.
The references are M. Goresky, R. Kottwitz and R. MacPherson, Purity of equivalued affine Springer fibers,
and T. Hikita, Affinene Springer Fibers of Type A and Combinatorics of Diagonal Coinvariants, arXiv:1203.5878
 Title: Slodowy slices
Abstract: This talk will be an effort to explain how Slodowy slices give information about
the structure of Springer fibers for loop groups and how the Slodowy slices relate to Walgebras.
The affine Springer fibers varieties seem to be the ones that control the link between loop groups
and Macdonald polynomials (generalizing the link between ordinary Springer fibers and the affine Hecke
algebra). Hopefully by that time I will have understood enough of Slodowy’s work to say something
coherent about the connections between these varieties, double affine Hecke algebras, and Walgebras.
Besides Slodowy, important references here include work of Premet on finite Walgebras and recent
papers of Arakawa.
Problems
I will try to formulate some problems as I go.
 Work out the qversion of the fundamental diagram, with Macdonald polynomials, following the sketch in my Banff lecture (also my Stanford lecture?)....
 Work out the representation theory of a central extension using the
Clifford theory applied to a normal subgroup.
References

V. Kac, Infinite dimensional Lie algebras, Third edition.
Cambridge University Press, Cambridge, 1990.
xxii+400 pp. ISBN: 0521372151; 0521466938,
MR1104219.

R. Steinberg,
Lectures notes on Chevalley groups,
Yale University, (1967).
 K. Nelsen and A. Ram,
KostkaFoulkes polynomials and Macdonald spherical functions ,
preprint 2003,
arXiv:0401298, Published in
Surveys in Combinatorics 2003 , C. Wensley ed.,
London Math. Soc. Lect. Notes 307 ,
Cambridge University Press, 2003.