April 19

Definition 1 Let P denote the collection of parabolic subgroups of the finite group G with split BN-pair. That is, define

P = {^{x}P_{I} : I ⊂ S, x ∈ N}

where P_{I} ≤ G denotes the standard parabolic subgroup BW_{I}B corresponding with I ⊂ S and W_{I} = ⟨I⟩ ≤ W.

Proposition 1 If P, Q ∈ P are such that P ∩ Q ∈ P, then

gP ≤ Q  iff  P ≤ Q and g ∈ Q.

Corollary 1 Each parabolic P ∈ P is conjugate to a unique standard parabolic P_{I} for some I ⊂ S.

Let P ∈ P. Then by Corollary 1, we have P = ^{x}P_{I} for unique I ⊂ S and some x ∈ N. If

P_{I} = L_{I}V_{I}

is the Levi decompositions of P_{I}, then we have

P = ^{x}P_{I} = ^{x}L_{I}^{x}V_{I}

is the Levi decomposition of P. Write L_{P} = ^{x}L_{I} and V_{P} = ^{x}V_{I}.

Suppose Q ∈ P is another parabolic subgroup. Then by Corollary 1, we have that Q = ^{y}P_{J} for some unique J ⊂ S and some y ∈ N. Moreover, P is not conjugate to Q unless I = J. However, L_{P} and L_{Q} can be conjugate even if I ≠ J.

Suppose, indeed, that I ≠ J, but P and Q have a common Levi subgroup L = L_{P} = L_{Q}. Then the operator R_{L}^{G} : ch (L) → ch (G) can be defined in this situation in two ways.

\xymatrix{ & ch (P) \ar[r]^{^{Ind^{G}^{P}}} & ch (G) \\ _{ch (L)} \ar@{-->}[ur]^{∼} \ar[dr]^{∼} \\  & _{ch (Q)} \ar[r]^{_{Ind_{G}_{Q}}} & _{ch (G)}}

Evidently R depends on the parabolic subgroup, so we will write R_{L,P}^{G} and T_{L,P}^{G} instead of R_{L}^{G} and T_{L}^{G} for the remainder of this lecture. We will show that R_{L,P}^{G} = R_{L,Q}^{G}, justifying the abbreviated notation.

Recall from last week the following version of the Mackey Intertwining Number Theorem for generalized induction.

Theorem 1 Let P, Q ∈ P with standard Levi subgroups L_{P} and L_{Q} and Weyl groups W_{L} and W_{M}. Let D be a set of double coset representatives of W_{I}\W/W_{J}. Let ξ ∈ ch (L) and η ∈ ch (M). Then

⟨R_{L,P}^{G} ξ, R_{M,Q}^{G}η⟩_{G} = ∑_{w∈D} ⟨T_{L∩^{w}M,L∩^{w}Q}^{L}ξ,^{w}T _{L^{w}∩M,P^{w}∩M}^{M}η⟩_{L∩^{w}M}

Theorem 2 Let P, Q ∈ P be parabolic subgroups having a common Levi subgroup L. Then

R_{L,P}^{G} λ = R_{L,Q}^{G}λ

for λ ∈ ch (L).

Proof. The claim is true for groups G with BN-rank 0 since in this case, G = B so that L = G is the only parabolic subgroup. Assume now that the claim is true for groups G with BN-rank < n. We have from Theorem 1 the following

⟨R_{L,P}^{G} λ, R_{L,P}^{G} λ⟩_{G} = ∑_{x∈D} ⟨T_{L∩^{x}L,L∩^{x}P}^{L} λ,^{x}T _{L^{x}∩L,P^{x}∩L}^{L}λ⟩_{L∩^{x}L}
⟨R_{L,P}^{G} λ, R_{L,Q}^{G}λ⟩_{G} = ∑_{x∈D} ⟨T_{L∩^{x}L,L∩^{x}Q}^{L}λ,^{x}T _{L^{x}∩L,P^{x}∩L}^{L}λ⟩_{L∩^{x}L}
⟨R_{L,Q}^{G}λ, R_{L,Q}^{G}λ⟩_{G} = ∑_{x∈D} ⟨T_{L∩^{x}L,L∩^{x}Q}^{L}λ,^{x}T _{L^{x}∩L,Q^{x}∩L}^{L}λ⟩_{L∩^{x}L} .

But all three are the same by the induction hypothesis. Therefore, we have

⟨R_{L,P}^{G} λ − R_{L,Q}^{G}λ, R_{L,P}^{G} λ − R_{L,Q}^{G}λ⟩_{G} = 0

so that R_{L,P}^{G} λ = R_{L,Q}^{G}λ.

Corollary 2 (Strong Conjugacy Theorem) Let I, J ⊂ S and assume L_{I} = ^{x}L_{J} for some x ∈ W. Let ψ ∈ ch (L_{J}) and let φ = ^{x}ψ ∈ ch (L_{I}). Then

R_{L_{I},P_{I}}^{G}φ = R_{L_{J} ,P_{J}}^{G} ψ.

Proof. We have that L_{I} = ^{x}L_{J} is the Levi subgroup for ^{x}P_{J} as well as for P_{I} so that by Corollary 2,

R_{L_{I},P_{I}}^{G}φ = R_{L_{I},^{x}P_{J}}^{G} φ = R_{^{x}L_{J} ,^{x}P_{J}}^{G}^{x}ψ = Ind_{^{x}P_{J}}^{G} ^{ x̃}ψ = Ind_{^{x}P_{J}}^{G}^{x}ψ̃ = Ind_{P_{J}}^{G} ψ̃ = R_{L_{J} ,P_{J}}^{G} ψ.

Recall the following definition.

Definition 2 A virtual character ξ ∈ ch (G) is called cuspidal if T_{L_{I}}^{G}ξ = 0 for all I ⊊ S.

Theorem 3 (Harish-Chandra) Let I, J ⊂ S and let φ and ψ be irreducible cuspidal characters of L_{I} and L_{J} respectively. Then

1. ⟨R_{L_{I}}^{G}φ, R_{L_{J}}^{G} ψ⟩ = 0 unless there exists x ∈ W with L_{I} = ^{x}L_{J} and φ = ^{x}ψ.

2. In the situation that ⟨R_{L_{I}}^{G}φ, R_{L_{J}}^{G} ψ⟩ ≠ 0, we have R_{L_{I}}^{G}φ = R_{L_{J}}^{G} ψ.