Presentation on theme: "Computer Calculations in the Unitary Dual Notes:"— Presentation transcript:
unitarity@home Computer Calculations in the Unitary Dual Notes: www.math.umd.edu/~jda/singapore
Let G be a real reductive group. Theorem (Vogan) There is a finite algorithm to compute the unitary dual of G. Note: This is much weaker than saying there is a finite algorithm for a single family of groups such as SL(n) or SO(n,C). Proposal: Write a computer program to compute the unitary dual of a fixed real group. Note: There is a big difference between a theorem asserting the existence of a finite algorithm, and a computer program. unitarity@home (mersenne@home, seti@home)
1. To tell us more about the unitary dual. 2. As a way to learn about the unitary dual and the technology involved. 3. Examples are fascinating; computers make more complicated examples accessible. 4. There are many interesting partial problems to be solved. It is not necessary to compute the full unitary dual to be successful. 5. It is fun. A lot of people will be involved (I hope). Best outcome: based on what we learn from the computer we can compute the answer “by hand”. Example: Model orbit in E 8 (Adams, Huang, Vogan), Non-arithmetic groups (Mostow, Deligne) Why do this?
Why not to do this? 1. Maybe we won’t understand about the unitary dual, even if we get the answer. 2. Poor substitute for thinking. 3. Generate lots of data which is hard to interpret. 4. Huge investment of time. 5. Likelihood of failure: it is not clear this can be done. 6. Can’t trust computers How do you know if the answer is correct? Maple test for positive definiteness It is very difficult to implement sgn(x) Example:
What is known: 1.GL(n) (R, C, H) (Vogan) 2.Complex classical groups (Barbasch) 3.SL(2,R), Sp(4,R), SO(5) (Bargmann, ?) 4.G 2 (Vogan) (including non-linear) 5.Spherical unitary dual of Sp(2n, R), SO(n,n), SO(n+1,n), F 4 (R) (Barbasch, unpublished; also p-adic case)
Sketch of proof of the Theorem: Uses the notion of K-character and Hermitian representation (see Vogan’s lecture) Standard module: I(G,P, )=I(P with irreducible quotient I(P, Derived functor module: R q ( ) (Huang’s lecture) Real infinitesimal character Huang’s lecture) Theorem (Knapp/Vogan) Suppose is an irreducible unitary representation of G. Then I(P, ) for some irreducible unitary representation with real infinitesimal character (readily computable).
Definition: (Salamanca, Vogan): An infinitesimal character is small if it is real and in the convex hull of W . An admissible representation is small if it has small infinitesimal character. Conjecture: (Salamanca, Vogan): Suppose is an irreducible unitary representation with real infinitesimal character. Then =R q ( ) where is small (q, are readily computable). Proposition: There is a finite algorithm to compute the small unitary representations. (The proof will be sketched below). By the Conjecture it is enough to compute the small unitary representations, and by the Proposition there is a finite algorithm to do this: Corollary: Assuming the conjecture, there is a finite algorithm to compute the unitary dual.
In the absence of the Corollary we still have the following version. Proposition * : For every group G there is a non-negative integer n(G) with the following property. Suppose is an irreducible unitary representation with real infinitesimal character. Then =R q ( ) where is an irreducible unitary representation with real infinitesimal character in the convex hull of n(G) . Corollary * : There is a finite algorithm to compute the unitary dual of G. Note: This version is essentially useless for computations. The proof of Proposition* is a variant of the proof of the Proposition, which we now sketch.
Sketch of the proof of the Proposition: (there is a finite algorithm to compute the small unitary representations): Basic point: 1. there is a finite number of minimal K-types to consider 2. for each a finite number of representations of G, 3. for each a finite number of K-types to check for unitarity. Definition: A K-type is unitarily small if its highest weight is in the convex hull of the weights of the exterior algebra of the weights of Lie(G)/Lie(K). Remark: In spite of its name, this is a fairly large set. Example: Sp(2n,C), is unitarily small iff it is in the convex hull of (2n,2n-2,…,2); pretty hopeless for computational purposes.
Proposition: Suppose is a small irreducible Hermitian representation. Then: 1.The lowest K-type of is unitarily small, 2. unitarity of is detected on the unitarily small K-types of Fix a unitarily small K-type Associated to is a parabolic subgroup P=MAN and a representation of M such that every irreducible representation with this lowest K-type is the unique irreducible quotient I(P, x ) of I(P, x ) for some. Lemma: Unitarity of I(P, x ) is constant on “reducibility” regions in the plane. There are only a finite number of such regions (with small infinitesimal character).
Example: Spherical unitary representations of Sp(4) The left hand picture shows the regions in question. The right hand picture shows the unitary dual of Sp(4,R) and Sp(4,F) for F p-adic. The picture for Sp(4,C) is the same with the line segments deleted (Lefschetz principle). Theorem: I(P, x ) is unitary if and only if it is unitary on all unitarily small K- types. Serious problem: find a smaller set of K-types which detect unitarity.
Summary: 1. Reduce to small (real) infinitesimal character. 2. Fix a unitarily small K-type. 3. Compute finite number of regions in the plane. 4. Pick a representative point in each such region. Compute the signature of the Hermitian form on every unitarily small K- type. Computational Ingredients: 1. Encoding of all of the objects: Cartan subgroups, standard modules, Weyl groups 2. Multiplicity of K-types in standard modules (“easy”) 3. Kazhdan-Lusztig algorithm, at all infinitesimal characters 4. Computation of Hermitian form on all unitarily small K-types
More details on the computational ingredients: 1. List the Cartan subgroups. Parmetrize characters of them. Proposal: use the parameters of [A, Bowdoin conference]: (x,T,B,y, d T, d B) T,B are a Cartan and a Borel subgroup in G, d T, d B similarly in v G y is in L G, y^2 in Z( v G) (for example…) Computing Weyl groups: (Vogan duality) Homework: Read and thoroughly understand Vogan duality (Irreducible Characters 4, Duke ?). Generalize to p-adic groups.
2. Multiplicity of K-types in standard modules This is in principal known: restriction from M to M intersect K. GL(n): O(n), O(p)xO(q) O(n), O(m 1 )xO(m 2 )…xO(m r ) Requires many “branching” type laws. 3. Kazhdan-Lusztig at any infinitesimal character reduces, via IC4 duality, to infinitesimal character . Even this is very hard. 4. Computing the Hermitian form on K-types: use the step algebra of Mikkelson.There is serious work to be done here to apply the step algebra, and to implement it computationally.
Spherical unitary representations of real, complex and p-adic groups (Vogan, Salamanca, Barbasch) X( )=irreducible spherical representation with (real) parameter Assume w 0 ( )=- (iff Hermitian) Theorem: For each K-type there is a Hermitian operator A( ) on Hom( X( )) such that X( ) is unitary if and only if A( ) is positive semi-definite. Theorem (Barbasch,Moy): In the p-adic case X( ) is unitary if and only if A( ) is positive semi-definite for all irreducible representations of W. Note: It is enough to take to be the regular representation: the positive definiteness of this size Order(W) matrix A( ) is equivalent to unitarity. Example/Test Case:
Consider real or complex groups. Let A( be the operator A( ) on the Weyl group representation on the 0-weight space of Theorem: X( ) is unitary if and only if A( is unitary for all unitarily small Problem: Compute the operator A( (this might be hard). There is a notion of “petite” representation (Vogan’s talk next week, possibly) for which A( is easy to compute. The petite representations come close to detecting unitarity. ( unitary impilies A( positive semi-definite for all petite , but not quite conversely. You get too many representations.) Important Problem: Find a nice set of K-types which is enough to detect unitarity in the real and complex cases (bigger than petite, smaller than the set of unitarily small).
Computations of Stembridge (and Adams) John Stembridge has written software to compute the spherical unitary dual of any split p-adic group, and also gives some information for real or complex groups, by computing the operator A. Complete answer: small classical groups, F 4 (p-adic) Should work for E 6, maybe E 7, not E 8 Good test case. Barbasch: independently has done F 4 (R, C, p-adic) Computational issues: rational arithmetic, computing positive definiteness, computing an explicit model of any irreducible representation of W
Example: The spherical unitary dual of F 4 over the p-adics (Stembridge, Adams, Barbasch) There are 284 facets in the plane. Precisely 59 of these are unitary. These are the union of the closure of 18 facets. There are 2, 0, 11, 3, 2 of these of dimension 4, 3, 2, 1, 0, respectively.
Closed Regions (old numbering/dimension): Closure of facet (59/4): (58/3),(33/2),(24/1),(9/0) Closure of facet (51/4): (54/3),(50/3),(21/2),(33/2),(30/2),(4/1),(11/1),(24/1),(9/0),(1/0) Closure of facet (56/2): (23/1),(31/1),(34/1),(25/0),(16/0),(1/0) Closure of facet (37/2): (7/1),(18/1),(10/0),(2/0) Closure of facet (55/2): (22/1),(34/1),(25/0),(16/0),(12/0) Closure of facet (53/2): (6/1),(23/1),(32/1),(16/0),(5/0),(1/0) Closure of facet (52/2): (22/1),(29/1),(32/1),(16/0),(12/0),(5/0) Closure of facet (57/2): (4/1),(31/1),(25/0),(1/0) Closure of facet (38/2): (7/1),(39/1),(27/0),(2/0) Closure of facet (49/2): (20/1),(29/1),(48/1),(19/0),(12/0),(5/0) Closure of facet (46/2): (7/1),(47/1),(28/0),(2/0) Closure of facet (43/2): (20/1),(42/1),(19/0),(27/0),(12/0) Closure of facet (45/2): (15/1),(44/1),(28/0),(14/0),(8/0) Closure of facet (36/1): (26/0),(13/0) Closure of facet (40/1): (13/0) Closure of facet (41/1): (26/0),(3/0) Closure of facet (35/0): Closure of facet (17/0):
1. Mathematica, Matlab, Maple John Stembridge has a Coxeter group package for Matla 2.LiE: free, Finite dimensional representations of complex classical Lie algebras 3.Magma: General algebra package, including character theory of finite groups, matrix groups over arbitrary rings 4.Gap: free, Group Algebra Package, finite groups, including the Atlas of finite groups and other goodies 5.Java, C++ 6.Perl: free, the “glue” Software:
** Construct the computational framework (encoding for standard representations, Weyl groups, orbits, …) 1. Representation theory library: Web based, compute lots of information about representations, dictionary, … Make an Atlas of Reductive Groups 2. Spherical unitary dual of p-adic exceptional groups (E 8 ) 3. Compute the operator A for general spherical representations of real groups (non-petite) 4. Kazhdan-Lusztig algorithm (Casselman, Kottwitz) 5. Regions in the plane (Stembridge) Some partial problems:
Conjecture: Let G be a simple split group over R, C or a p-adic field. Assume one root length. Let G ~ be an n-fold cover of G. Then the spherical unitary dual of G and the quasi-spherical unitary dual of G ~ are the same when scaled by a factor of n. Examples: GL(n) (Huang), G 2 (R) (Vogan), GL(n,C) (Tadic) Perhaps it is possible to prove this in some cases by looking at the computation, without explicitly computing the unitarity set for G or G ~