Lie Theory without groups 2020 Erd s Memorial Lecture Fall Western - - PowerPoint PPT Presentation

Lie Theory without groups

2020 Erdős Memorial Lecture Fall Western Sectional Meeting, October 24 Andrei Okounkov

Lie Group = a group + a manifold

SL(n) SO(n) Sp(n) exceptional

Lie groups are everywhere !

The world of Lie groups has been explored and inhabited ….. Lie groups are everywhere !

….. and many Lie theorists have been searching for new worlds.

Many directions of this search have been strongly influenced by applications in mathematical physics (supersymmetric quantum gauge and string theories, in particular) and algebraic geometry, especially to enumerative geometry of curves and sheaves. In this talk, I want to motivate them from the point of view of representation theory and special functions. I hope this will appeal to those with interests very distant from core Lie theory. First a brief summary of classical theory :)

… . .

representation theory goes hand–in-hand with harmonic analysis on homogeneous spaces and related special functions For instance, if we want to talk about functions on the sphere (including, omg, the spherical harmonics!) we note that:

representation theory goes hand–in-hand with harmonic analysis on homogeneous spaces and related special functions For instance, if we want to talk about functions on the sphere (including, omg, the spherical harmonics!) we note that:

• S2 = SU(2)/H, where H = diag(eit,e-it) ;

representation theory goes hand–in-hand with harmonic analysis on homogeneous spaces and related special functions For instance, if we want to talk about functions on the sphere (including, omg, the spherical harmonics!) we note that:

• S2 = SU(2)/H, where H = diag(eit,e-it) ;
• functions on SU(2) are matrix

coefficients of irreducibles ;

representation theory goes hand–in-hand with harmonic analysis on homogeneous spaces and related special functions For instance, if we want to talk about functions on the sphere (including, omg, the spherical harmonics!) we note that:

• S2 = SU(2)/H, where H = diag(eit,e-it) ;
• functions on SU(2) are matrix

coefficients of irreducibles ;

• irreducibles for SU(2) are polynomials of

fixed degree d in x and y ;

representation theory goes hand–in-hand with harmonic analysis on homogeneous spaces and related special functions For instance, if we want to talk about functions on the sphere (including, omg, the spherical harmonics!) we note that:

• S2 = SU(2)/H, where H = diag(eit,e-it) ;
• functions on SU(2) are matrix

coefficients of irreducibles ;

• irreducibles for SU(2) are polynomials of

fixed degree d in x and y ;

• monomials (xy)d/2 are H-invariant and

corresponding matrix elements are functions on S2 .

On the n-dimensional sphere Sn, we may be looking e.g. for radial eigenfunctions of the Laplace operator Δ. Radial means a function on H\G/H, where G=SO(n+1), H=SO(n).

z=exp(iθ)

These have the form z-k F(z2,a,b) where

• a=(n-1)/2 encodes the dimension
• -b=0,1,2,….. encodes the number of the

number of the harmonic

• F is the a+c=b+1 special case of the Gauss

hypergeometric function

• the second order DE satisfied by F generalizes the radial

part of the Laplace operator. Interpolates in dimension, in particular !

• the second order DE satisfied by F generalizes the radial

part of the Laplace operator. Interpolates in dimension, in particular !

• b = 0,1,2, ….. makes the series terminate. Nonterminating

cases correspond to ထ-dimensional representations

• the second order DE satisfied by F generalizes the radial

part of the Laplace operator. Interpolates in dimension, in particular !

• b = 0,1,2, ….. makes the series terminate. Nonterminating

cases correspond to ထ-dimensional representations

• it is remarkable that there is an explicit formula for F, and

also explicit integral representations, formulas for the value at x=1, formulas for the monodromy in x, formulas for commuting difference equations in a, b, c, et cetera. All have representation-theoretic meaning, proofs, and applications.

Gauss hypergeometric function is the rank 1 case of the multivariate Jacobi polynomials associated to an arbitrary root system by G.Heckman and E.Opdam. The q-difference extension of this theory, initiated by I.G.Macdonald and transformed by I.Cherednik, covers also p-adic special functions and has found applications all over mathematics, including e.g. many applications to interacting particle systems and other classical problems

• f probability theory.

The hypergeometric equation is also the simplest instance of the Knizhnik- Zamolodchikov differential equations from Conformal Field Theory. The q-difference deformation of KZ equations, introduced by I.Frenkel and N. Reshetikhin a few years later, plays an equally important role in the analysis

• f 2-dimensional lattice models before one

takes the continuum limit.

While very general, and encompassing a wide range of applications, both Macdonald-Cherednik and q-Knizhnik-Zamolodchikov equations are still “rooted” in classical Lie theory, with root systems etc. imprinted in vector spaces in which the difference operators act, the structure of the singularities, etc.

It is certainly important to study general linear differential and difference equations in the context of Riemann-Hilbert type monodromy problems, general properties of D-modules, et cetera It is also important to study general associative algebras in place of universal enveloping algebras ( group-invariant differential operators on a Lie group G)

….. but it may be difficult to find habitable places where several points of view meet and create a fertile environment

Remarkably, there is whole galaxy of new of new possibilities where modern high energy physics meets representation theory and algebraic geometry

Any great idea can always be traced to a number of sources, but for me, personally, a very important moment happened back in 2007 or 2008 when I first heard from Nekrasov and Shatashvili that all special functions above should be special cases of functions that count holomorphic maps from a Riemann surface C to certain special algebraic varieties X.

X C

f

In algebraic geometry, it is very interesting to count curves in some X satisfying such and such geometric

• conditions. For instance, there are 12

rational cubics through 8 points in the plane, and 2875 lines on a quintic threefold.

X=the parameter space for vacua, i.e. lowest energy states

space f

In mathematical physics, we can model states of a very large system as modulated vacuum, that is, near vacuum whose parameters (e.g. temperature and pressure) vary in space With supersymmetry, counts like 12 or 2875 are then interpreted as indices

• f certain evolution operators

(infinite-dimensional versions of the Dirac operator). These are important invariants of continuous deformations, e.g. scale transformation.

space f

An index is really a vector space, and it carries the representation of the symmetry groups of both the source and the target It is also graded by discrete invariants (=degree) of the map f and it is convenient to encode them as a representation of a “Kähler torus”. (The two kinds of variables are exhanged by a remarkable symmetry first observed by Intrillegator and Seiberg)

X

Xv

Our q-hypergeometric friend appears in, probably, the simplest possible situation when

• the source C=P1=Riemann sphere

with automorphism q \in C*

• the target X=

=T*P1, with similar automorphism a and also scaling \hbar of the cotangent direction

• the Kähler variable z counts the degree of the map f: P1 -> T*P1
• the map is constrained to something like f(\infty) = \infty

T*P1 , which the resolution of singularities of the usual cone = nilpotent cone of sl(2), is the unicellular organism of flora of possible X The way these cells fit together in a general X may be described by certain hyperplane arrangements, which also reflect the singularities in the Kähler and equivariant

• variables. In total, two root-like systems, of

different ranks, exchanged by the duality a<->z (for which T*P1 is self-dual). blow-up the singularity is more fundamental, resolution is way to handle it

“the anatomy of a Lie algebra”

For a general simple Lie algebra g, the corresponding singularity is the nilpotent cone in g (equivalently, its dual g*) Other conjugacy classes form the family of deformations of the nilpotent cone as an algebraic symplectic variety. Closely related to symplectic leaves of the Lie- Kirillov-Kostant Poisson structure on g*

It is a very interesting question which singularities can appear in the moduli spaces of vacua in interest. There is a big intersection with the equivariant symplectic resolutions X0, studied in depth by D. Kaledin, Y. Namikawa, and many others. These satisfy

• There is a resolution of singularities f: X—>X0, with X symplectic.
• There is a group action that scales the symplectic form and

contracts X0 to a point. It may be useful to separate the global properties (such as being globally a cone) from the local ones. In physics, global properties of the Coulomb branch depend on the topology of the space-time of but local properties are the same.

x 3 x 3 A very important new example is Hilbert scheme of points of C2, which resolves the symmetric product SnC2. In the Hilbert scheme, we keep track of the ideal of all polynomials that vanish at a given collection

• f points

Many more examples among Nakajima quiver varieties.

Curve counting functions (“vertex functions”) for the Hilbert scheme

• are like the hypergeometric function is which the constant term is

the Macdonald polynomial,

• are the basic building blocks for both the Donaldson-Thomas and

Gromov-Witten counts of curves in algebraic 3-folds Like T*P1, Hilb(C2) is self-dual. In general, these vertex functions are under excellent representation theoretic-control, using algebraic structures that generalize both Cherednik’s double affine Hecke algebras and the quantum groups on which the qKZ equations rest. One can say that the Nekrasov- Shatashvili vision did come true thanks to the work of many people including M. Aganagic, R. Bezrukavnikov, D. Maulik, A. Smirnov ….. and my humble self.

https://www.math.columbia.edu/~okounkov/courses.html also here

f

The link to actual representation theory is as tight as it was in harmonic analysis: following the insights of Gaiotto, Braverman-Finkelberg-Nakajima, ….. a quantization of Xv acts on maps to X by correspondences The actual enumerations is a refined character of this representation, in the spirit of Nekrasov’s qq-characters of representation of quantum groups

X

Xv

At the end of the talk, I’d like to discuss the following question. Locally the Lie(G) looks like G, and the spaces X discussed so far are meant to generalize that local behavior. What is then the difference between the Lie group and the Lie algebra for us ?

I think the difference is in how the different symplectic leaves fit together globally, and more precisely, in what parametrizes leaves in the boundary of a given leaf A guiding example may be Hilb(P2), where leaves are parametrized by points that got

• nto the zero locus of the Poisson structure,

and that can be a elliptic curve, a rational cubic, or e.g. a triple line, like it is for Hilb(C2)