Software Project: Weakly Fair A_{𝔮}(λ)

August 17, 2006

This is a project recommended by David Vogan at the Atlas workshop, July 2006.

Suppose 𝔮 = 𝔩⊕𝔲 is a θ-stable parabolic subalgebra of 𝔤, and 𝔥 is a Cartan subalgebra of 𝔩. Suppose λ ∈ 𝔥^{∗} defines a one-dimensional representation of 𝔩, i.e. ⟨λ, α^{∨}⟩ = 0 for all roots of 𝔥 in 𝔩. Fix Δ^{+} = Δ^{+}(𝔥, 𝔤) containing Δ(𝔥, 𝔲), and let ρ = \frac{1}{2} ∑_{α∈Δ^{+}} α and ρ(𝔲) = \frac{1}{2} ∑_{α∈Δ(𝔥,𝔲)} α as usual. We normalize cohomological induction so A_{𝔮}(λ) has infinitesimal character λ + ρ.

We say λ is good if

(1)  ⟨λ, α^{∨}⟩ > 0 for all α ∈ Δ^{+}

and is weakly fair if

(2)  ⟨λ + ρ(𝔲), α^{∨}⟩ ≥ 0 for all α ∈ Δ^{+}

If λ is good then A_{𝔮}(λ) is non-zero, irreducible, unitary, and has regular integral infinitesimal character. These representations are quite well understood. If λ is weakly fair then A_{𝔮}(λ) is non-zero, unitary, but not necessarily irreducible. It is of interested to compute this sum: some of these are interesting unitary representations.

That is write

(3)  A_{𝔮}(λ) = ∑_{i} a_{i}π_{i}

where each π_{i} is irreducible, and a_{i} ∈ ℕ.

Problem: Use the atlas software to compute (3).

Here is a sketch of what is involved.

Fix 𝔮 = 𝔩 ⊕ 𝔲 and weakly fair λ.

This is all going on in the block of the trivial representation, so label the parameters in this block 0, …, n. Write π(i) for the irreducible representation with parameter i, and I(i) for the standard representation.

Recall A_{𝔮}(0) has infinitesimal character ρ and is unitary. It occurs in the output of the blocku command. Find i (in the output of blockd or blocku) so that A_{𝔮}(0) = π(i).

Now choose w ∈ W so that

(4)  ⟨w(λ + ρ), α^{∨}⟩ ≥ 0 for all α ∈ Δ^{+}.

Next compute

(5)  w^{−1} · π(i) = ∑_{j∈S} c_{j}π(j)

where S ⊂ {0, …, n}. Here w^{−1} · π(i) is the coherent continuation action. The wgraph command gives the information needed to compute this action. This is carried out by the helper application coherentContinuation

at www.liegroups.org/software/helpers (at some point this will be built

into atlas itself).

Now list the simple roots β_{1}, …, β_{r} such that

(6)  ⟨w(λ + ρ), β_{k}^{∨}⟩ = 0.

Now discard those π(j)′s for which some β_{k} is in the τ-invariant of π(j). That is write S′ ⊂ S for the set of j ∈ S for which

(7)  for all 1 ≤ k ≤ r, β_{k} is not in the τ invariant of π(j)

This is available from the output of the block command, see the Appendix. We discard π(j) if:

(8)  some β_{k} is of type ic, r1, r2, or C − for π(j).

Then we have

(9)  A_{𝔮}(λ) = ∑_{j∈S′} c_{j}π′(j)

where

(10)  π′(j) = ψ_{w(λ+ρ)}^{ρ}(π(j))

the translation of π(j) at infinitesimal character ρ to infinitesimal character λ + ρ. Each π′(j) is irreducible, and (by the assumption on the τ-invariant) non-zero.

1 Appendix

We take the opportunity to summarize in one place information about types of roots. Fix a block with parameters 0 ≤ i ≤ n. Recall the cross action is defined on parameters; we write w × i = j.

For each parameter i each simple root is listed by type in the output of

block:

i1: α is imaginary, noncompact, and type I, meaning the following equivalent conditions hold:

(a) s_{α} ∉ W (G(ℝ), H(ℝ))

(b) the Cayley transform c_{α} is single-valued

(c) s_{α} × i ≠ i

(d) this is an SL(2, ℝ)-situation

(e) c_{α}(α) is of type r1 i2: α is imaginary, noncompact, and type II, meaning the following equiv-

alent conditions hold:

(a) s_{α} ∈ W (G(ℝ), H(ℝ))

(b) the Cayley transform c_{α} is double-valued

(c) s_{α} × i = i

(d) this is a P GL(2, ℝ) situation

(e) c_{α}(α) is of type r2 r1: α is real, satisfies the parity condition, and is type 1, meaing the fol-

lowing equivalent conditions hold:

(a) α(h) ≠ −1 for any h ∈ H(ℝ),

(b) the Cayley transform c^{α} is double valued

(c) s_{α} × i = i

(d) this is an SL(2, ℝ)-situation

(e) c^{α}(α) is of type i1 r2: α is real, satisfies the parity condition, and is of type 2, meaing the

following equivalent conditions hold:

(a) α(h) = 1 for some h ∈ H(ℝ),

(b) the Cayley transform c^{α} is single valued

(c) s_{α} × i ≠ i

(d) this is a P GL(2, ℝ)-situation

(e) c^{α}(α) is of type i2 rn: α is real, and does not satisfy the parity condition. ic: α is imaginary and compact, C+: α is complex, and θ(α) > 0 C-: α is negative, and θ(α) < 0

The roots in the τ-invariant of the irreducible representation π(i) with parameter i are determined as follows. If α is simple then:

1. α imaginary: α ∈ τ(π(i)) ⇔ α is compact

2. α real: α ∈ τ(π(i)) ⇔ α satisfies the parity condition

3. α complex: α ∈ τ(π(i)) ⇔ θ(α) < 0 In the notation of atlas this means

(11)  \begin{matrix}τ(π(i)) : & ic, r1, r2, C− \\ not in τ(π(i)) : & i1, i2, rn, C+ \end{matrix}