Lie Theory Workshop
Fall 2015 meeting of Lie Groups, Lie Algebras, and their Representations
Louisiana State University • November 7–8, 2015
All talks will be held in Lockett Hall rooms 277 or 285.
Saturday, November 7
9:30am–10:00am
Coffee
10:00am–11:00am
Joseph Wolf (Berkeley)
Real Lie Groups and Complex Flag Manifolds
Let $G$ be a complex simple direct limit group. Let $G_{\Bbb R}$ be a real form of $G$ that corresponds to an hermitian symmetric space. I'll describe the corresponding bounded symmetric domain in the context of the Borel embedding, Cayley transforms, and the Bergman-Shilov boundary. Let $Q$ be a parabolic subgroup of $G$. In finite dimensions this means that $G/Q$ is a complex projective variety, or equivalently has a Kaehler metric invariant under a maximal compact subgroup of $G$. Then I'll show just how the bounded symmetric domains describe cycle spaces for open $G_{\Bbb R}$ orbits on $G/Q$. These cycle spaces include the complex bounded symmetric domains. In finite dimensions they are tightly related to moduli spaces for compact Kaehler manifolds and to representations of semisimple Lie groups; in infinite dimensions there are more problems than answers. Finally, time permitting, I'll indicate how some of this goes over to real and to quaternionic bounded symmetric domains.
Let $G$ be a complex simple direct limit group. Let $G_{\Bbb R}$ be a real form of $G$ that corresponds to an hermitian symmetric space. I'll describe the corresponding bounded symmetric domain in the context of the Borel embedding, Cayley transforms, and the Bergman-Shilov boundary. Let $Q$ be a parabolic subgroup of $G$. In finite dimensions this means that $G/Q$ is a complex projective variety, or equivalently has a Kaehler metric invariant under a maximal compact subgroup of $G$. Then I'll show just how the bounded symmetric domains describe cycle spaces for open $G_{\Bbb R}$ orbits on $G/Q$. These cycle spaces include the complex bounded symmetric domains. In finite dimensions they are tightly related to moduli spaces for compact Kaehler manifolds and to representations of semisimple Lie groups; in infinite dimensions there are more problems than answers. Finally, time permitting, I'll indicate how some of this goes over to real and to quaternionic bounded symmetric domains.
11:15am–12:15pm
Raul Gomez (Cornell)
Invariant trilinear forms on induced representations of real rank one groups (joint work with B. Speh)
Bernstein and Reznikov introduced a triple integral formula to describe a family of invariant trilinear forms for induced representations of $PGL(2, \mathbb{R})$. However, they left open the question of computing the full space of invariants. Using the definition of the Schwartz space of a Nash manifold, together with some homological algebra, we will show how to describe the remaining trilinear forms. We will then show how these results can be extended to induced representations of real rank one groups, refining in this way some results previously obtained by Clerc, Kobayashi, Ørsted and Pevzner.
Bernstein and Reznikov introduced a triple integral formula to describe a family of invariant trilinear forms for induced representations of $PGL(2, \mathbb{R})$. However, they left open the question of computing the full space of invariants. Using the definition of the Schwartz space of a Nash manifold, together with some homological algebra, we will show how to describe the remaining trilinear forms. We will then show how these results can be extended to induced representations of real rank one groups, refining in this way some results previously obtained by Clerc, Kobayashi, Ørsted and Pevzner.
12:15pm–2:00pm
Lunch
2:00pm–3:00pm
Ben Harris (Simon's Rock)
Regular Elliptic Discrete Spectra
Harmonic analysis on the unit circle involves writing a function as an infinite direct sum while harmonic analysis on the real line involves writing a function as a continuous integral. The first decomposition is discrete while the second is continuous. Philosophically, this is because the first group is compact and the second group is non-compact. In general, we say that an element of a Lie group is elliptic if it is contained in a compact subgroup. In this talk, we will survey old and new results linking elliptic elements with the existence of certain types of discrete spectra in abstract harmonic analysis.
Harmonic analysis on the unit circle involves writing a function as an infinite direct sum while harmonic analysis on the real line involves writing a function as a continuous integral. The first decomposition is discrete while the second is continuous. Philosophically, this is because the first group is compact and the second group is non-compact. In general, we say that an element of a Lie group is elliptic if it is contained in a compact subgroup. In this talk, we will survey old and new results linking elliptic elements with the existence of certain types of discrete spectra in abstract harmonic analysis.
3:15pm–4:15am
Hadi Salmasian (Ottawa)
The Capelli problem and spectrum of invariant differential operators
The Capelli identity is a mysterious result in classical invariant theory with a long history. It was demystified by Roger Howe, who used it in an ingenious and elegant fashion in the modern theory of representations of real reductive groups. In this talk, I will introduce the Capelli identity, and exhibit the relationship between an extension of this identity with certain polynomials which describe the spectrum of invariant differential operators on symmetric superspaces. These polynomials are analogs of the Jack and Knop-Sahi/Okounkov-Olshanski polynomials. This talk is based on a joint project with Siddhartha Sahi.
The Capelli identity is a mysterious result in classical invariant theory with a long history. It was demystified by Roger Howe, who used it in an ingenious and elegant fashion in the modern theory of representations of real reductive groups. In this talk, I will introduce the Capelli identity, and exhibit the relationship between an extension of this identity with certain polynomials which describe the spectrum of invariant differential operators on symmetric superspaces. These polynomials are analogs of the Jack and Knop-Sahi/Okounkov-Olshanski polynomials. This talk is based on a joint project with Siddhartha Sahi.
4:30pm–5:30pm
Tsao-Hsien Chen (Northwestern)
Springer theory for symmetric spaces
We consider the Springer correspondence in the case of symmetric spaces. In this setting various new phenomena occur which are not present in the classical Springer theory. For example, we obtain representations of (the Tits extension) of the braid group rather than just Weyl group representations. These representations come from cohomology of families of Hessenberg varieties. In the situations we consider the Hessenberg varieties can be interpreted as classical objects in algebraic geometry: Jacobians and moduli spaces of vector bundles on curves with extra structure, Fano varieties of k-planes in the intersection of two quadrics, etc. This is joint work with Kari Vilonen and Ting Xue.
We consider the Springer correspondence in the case of symmetric spaces. In this setting various new phenomena occur which are not present in the classical Springer theory. For example, we obtain representations of (the Tits extension) of the braid group rather than just Weyl group representations. These representations come from cohomology of families of Hessenberg varieties. In the situations we consider the Hessenberg varieties can be interpreted as classical objects in algebraic geometry: Jacobians and moduli spaces of vector bundles on curves with extra structure, Fano varieties of k-planes in the intersection of two quadrics, etc. This is joint work with Kari Vilonen and Ting Xue.
Sunday, November 8
8:30am–9:00am
Coffee
9:00am–10:00am
Cheng-Chiang Tsai (MIT)
Stratification of affine Springer fibers
We describe an inductive stratification for arbitrary affine Springer fibers. Each stratum is an iterated étale locally trivial fiber bundle over a slight generalization of affine Springer fibers for smaller groups. If time permits, we will describe some examples of the fibers of these fiber bundles, and discuss further expectations.
We describe an inductive stratification for arbitrary affine Springer fibers. Each stratum is an iterated étale locally trivial fiber bundle over a slight generalization of affine Springer fibers for smaller groups. If time permits, we will describe some examples of the fibers of these fiber bundles, and discuss further expectations.
10:10am–11:10am
Sean Rostami (Wisconsin)
On Fixers of Stable Functionals
The epipelagic representations of Reeder-Yu, a generalization of the “simple supercuspidals” of Gross-Reeder, are certain low-depth supercuspidal representations of reductive algebraic groups $G$. From a linear functional $f$ (on a vector space $V$ coming from a Moy-Prasad filtration) which is stable in the sense of Geometric Invariant Theory, such a representation can be constructed. It is known that these representations are compactly induced from the fixer in $G$ of $f$ and it is important to identify all the elements that belong to this fixer in as explicit and familiar a way as possible. In the situation of Gross-Reeder, there is a uniform answer, which I finished in summer 2015. In the more general case, qualitatively different answers can occur and interesting objects (e.g. the Legendre symbol) appear, even when the context (i.e. $G$ and $V$) is fixed. This work is in-progress.
The epipelagic representations of Reeder-Yu, a generalization of the “simple supercuspidals” of Gross-Reeder, are certain low-depth supercuspidal representations of reductive algebraic groups $G$. From a linear functional $f$ (on a vector space $V$ coming from a Moy-Prasad filtration) which is stable in the sense of Geometric Invariant Theory, such a representation can be constructed. It is known that these representations are compactly induced from the fixer in $G$ of $f$ and it is important to identify all the elements that belong to this fixer in as explicit and familiar a way as possible. In the situation of Gross-Reeder, there is a uniform answer, which I finished in summer 2015. In the more general case, qualitatively different answers can occur and interesting objects (e.g. the Legendre symbol) appear, even when the context (i.e. $G$ and $V$) is fixed. This work is in-progress.
11:30am–12:30pm
Peter Fiebig (Erlangen-Nürnberg)
Intersection matrices and modular representation theory
We review the approach of Andersen, Jantzen and Soergel to Lusztig's conjecture on the irreducible rational characters of reductive algebraic groups. Then we show that the problem of the degeneration of categorical structures for “small” primes cumulates in the problem of understanding a certain matrix constructed in terms of the root system.
We review the approach of Andersen, Jantzen and Soergel to Lusztig's conjecture on the irreducible rational characters of reductive algebraic groups. Then we show that the problem of the degeneration of categorical structures for “small” primes cumulates in the problem of understanding a certain matrix constructed in terms of the root system.
12:40pm–1:40pm
Christopher Bremer (Baton Rouge)
