Are there any dominant pivotal functors such that the regular representation is not mapped onto a multiple of the regular representation?

This question is related to Pivotal functors of that are substantially different from finite group homomorphisms. A tensor functor F:C→D is called dominant (sometimes called “surjective”) if for any Y:D, there is an X:C such that Y is a subobject of FX. It is known (“On fusion categories” by Pavel Etingof, Dmitri Nikshych, and Viktor … Read more

Which tensor power of a given representation contains the trivial one?

If R is an irreducible representation of a simple Lie-groups G I assume there is always a lowest integer n such that the tensor product representation R⊗R⊗…⊗R (n times) contains the trivial (or singlet) representation. I can more or less obtain n on a case-by-case basis (for example with G=SU(N) and R=Fund it appears n=N) … Read more

When is the unitary dual of a lscs group uniformizable?

Let G be a locally compact, second countable group. We equip the unitary dual ˆG with the Fell topology. I am looking for conditions which guarantee that the topological space ˆG is uniformizable. Here, a topological space X is called uniformizable if there exists a uniform space whose underlying topological space is homeomorphic to X. … Read more

decomposition into irreducible unitary representations: references for explicit formulas?

I’m looking for references of the decomposition of L2(Γ∖G), where G is a connected Lie group, and Γ⊂G a discrete lattice; for simplicity one may assume that G is the real point associated to a linear algebraic group defined over Q, without characters defined over Q, and Γ is an arithmetic/congruence lattice in G. Write … Read more

comprehensive presentation of the unitary dual of SO0(n,1)SO_0(n,1)

The unitary dual (unitary irreducible represenations) is determined for every connected noncompact semisimple Lie group of real rank one. I would like to have a reference for the particular case SO0(n,1). I know the paper of Baldoni Silva-Barbasch (“The unitary spectrum for real rank one”, Invent. math. 72) and the reference therein, but I would … Read more

Unitary irreps of the Poincare group in dimension <4

It is well-known that long ago, Wigner classified the unitary irreducible representations of the Poincare group in dimension 4. I am looking for a convenient reference describing all unitary irreducible representations of the Poincare group in dimensions 2 and 3. (I know how it can be done in principle. But I am looking for a … Read more

Uniform Roe algebra of virtually abelian group is type I C*-algebra?

Let G be an arbitrary (discrete) group. It acts by left translation on ℓ∞(G). The uniform Roe algebra of G is defined as the crossed product ℓ∞(G)⋊. Elmar Thoma has shown (Thoma, E., Eine Charakterisierung diskreter Gruppen vom Typ I, Invent. Math. 6, 190-196 (1968). ZBL0169.03802.) that the reduced group C^*-algebra C_{\mathrm{red}}^*(G) of a group … Read more

non unitary representations

A paper by M.L. Whippman in Rep. Math. Phys. 5 (1974), 81, mentions at the bottom of the second page ”the possible occurrence of non-unitary representations that arise when reducing a direct product of representations of a non-compact group”. I’d like to see a simple example of this situation. The context is unitary representations, but … Read more

Clebsch–Gordan decomposition for $\mathrm{SU}(2)$, in indices

Let $\pi_m$, $m \geq 0$, be the unitary irreps of $\mathrm{SU}(2)$. The Clebsch–Gordan decomposition then gives that $$ \pi_m \otimes \pi_n = \bigoplus_{k=0}^{\min(m,n)}\pi_{m+n-2k}.$$ But suppose I want to think of this decomposition as matrices. Evaluating at a point $x \in \mathrm{SU}(2)$, on the left I have $$ (\pi_m(x))_{ij} (\pi_n(x))_{pq}.$$ How do the indices $i$, $j$ … Read more

Supercuspidal with Iwahori fixed vector

Let $F$ be a local field. Is there a reference for the following fact: No supercuspidal representation of $GL_2(F)$ has an Iwahori-fixed vector? I have a proof, by I’d prefer a reference, because it is not enlightening. Rough sketch of proof: We can easily see that Iwahori-fixed vector implies depth-zero and that depth-zero supercuspidal are … Read more