2 Representations, characters and quasi-characters
Background of Thesis Ch. 2–4: tempered representations, the multiplicity \(m(\pi )\), Harish-Chandra characters, quasi-characters and their germs.
2.1 Representations and the multiplicity
A strongly continuous unitary representation of a topological group on a complex Hilbert space (Thesis §3.1).
\(\pi \) is irreducible and tempered: irreducible and weakly contained in the left regular representation of \(G\) (Thesis Ch. 1, §3.1.2). The Thesis uses the growth condition on matrix coefficients and notes the equivalence (Cowling–Haagerup–Howe); the weak-containment form is used here because it stays valid when the centre of \(G\) is noncompact.
For a unitary representation \(\pi \) of \(G(F)\),
where \(\pi ^\infty \) is the space of smooth vectors (Thesis §5.5).
For a continuous unitary representation of a topological group with no closed invariant subspaces other than \(0\) and the whole space, every bounded operator commuting with the group is a scalar.
Assumed from: [ 6 ] , 13.1.4 (i)\(\Rightarrow \)(ii).
In an irreducible unitary representation of a topological group, the centre acts by a character (Thesis §3.1.2).
A central element’s operator commutes with the whole group, so it is a scalar by Lemma 27.
The Haar measure of \(G(F)=\mathrm{GSpin}(W)(F)\times \mathrm{GSpin}(V)(F)\) (and of every inner form) is right invariant.
The modular character \(\Delta \) is a homomorphism to \(\mathbb {R}_{{\gt}0}\), trivial on the centre and invariant under continuous involutive automorphisms. Conjugation by an anisotropic vector \(w\) on one factor is a continuous involutive automorphism sending \(uw\) to \(q(u)q(w)(uw)^{-1}\), so \(\Delta (uw)^2=1\) and \(\Delta (uw)=1\). Scalars and products \(uw\) of anisotropic vectors generate \(\mathrm{GSpin}(U)(F)\) (Cartan–Dieudonné), so \(\Delta =1\). (The thesis takes unimodularity as standard, §2.1.)
Let \(S\hookrightarrow \Gamma \) be a closed embedding of unimodular locally compact, Hausdorff, second countable groups with Haar measures \(\mu ,\nu \). There is a measure \(m\) on \(S\backslash \Gamma \) with \(\int _\Gamma \varphi \, d\nu =\int _{S\backslash \Gamma }\int _S\varphi (sq)\, d\mu (s)\, dm(q)\) for all \(\varphi \in C_c(\Gamma )\), unique among the measures finite on compact sets (the Radon measures). (Used for \(H(F)/A_{G,H}(F)\subset G(F)/A_G(F)\), Thesis §7.1.)
Assumed from: Bourbaki, Intégration VII §2; [ 8 ] , Thm. 2.49.
The measure \(m\) of Literature 30 is unique among all Borel measures.
Any measure satisfying the formula is finite on compact sets: lift a compact set to \(\Gamma \) and test the formula on a bump function that is \(1\) on a neighbourhood of the lift.
Let \(G\) be a locally compact, Hausdorff, second countable group, \(P\) a closed subgroup and \(K\) a compact subgroup with \(G=PK\). For Haar measures \(dx,dp,dk\) there is \(c{\gt}0\) with \(\int _G f(x)\, dx = c\int _K\int _P f(pk)\, dp\, dk\) for continuous compactly supported \(f\).
Assumed from: [ 13 ] , Thm. 8.32; Bourbaki, Intégration VII §2; Casselman’s notes §3.1.
2.2 Characters and quasi-characters
\(D^G(x)=|\det (1-\mathrm{Ad}x)|_{\mathfrak g/\mathfrak g_x}|_F\) (Thesis §2.1), computed as the absolute value of the trailing coefficient of the characteristic polynomial of \(1-\mathrm{Ad}x\) on \(\mathfrak g\).
For \(x\in G(F)\) write \(\det (t-1+\mathrm{Ad}(x))=D(x)t^\ell +\dots \) (terms of higher degree), \(\ell =\operatorname {rank}G\). Then \(G'=\{ x: D(x)\neq 0\} \) is open and dense in \(G(F)\) and its complement has Haar measure zero.
Assumed from: [ 10 ] , Part V §3, p. 54 (also Introduction, p. 2), for \(G\) connected reductive; here \(G=\mathrm{GSpin}(W)\times \mathrm{GSpin}(V)\), \(\ell =(\lfloor m/2\rfloor +1)+(\lfloor (m+2r+1)/2\rfloor +1)\).
Over an algebraically closed field of characteristic \(0\), for \(n\ge 3\) and \(X\in \mathfrak {so}_n\), \(\dim \mathfrak {so}_n(X)\ge \lfloor n/2\rfloor \).
Assumed from: [ 12 ] , §7.11, Remark, and §7.13.
Over an algebraically closed field of characteristic \(0\), for \(n\ge 3\), a nilpotent \(X\in \mathfrak {so}_n\) whose centralizer has minimal dimension among nilpotent elements has partition \([n]\) (\(n\) odd) or \([n-1,1]\) (\(n\) even): \(X^{n-1}\neq 0\), resp. \(X^{n-2}\neq 0=X^{n-1}\).
Assumed from: [ 12 ] , §6.3 (regular \(\Leftrightarrow \) minimal centralizer dimension; partitions of the regular orbit).
For \(x\in G_{\mathrm{reg}}(F)\), \(\dim \mathfrak g_x=\operatorname {rank}\mathfrak g=\min _{X\in \mathfrak g}\dim \mathfrak g_X\).
Both sides split over \(\mathfrak g=\mathfrak {gspin}(W)\times \mathfrak {gspin}(V)\). For \(g\in \mathrm{GSpin}(U)\), \(\mathfrak {gspin}(U)_g\cong F\oplus \mathfrak {so}(U)_{P(g)}\); if \(g\) is semisimple with \(\mathfrak {gspin}(U)_g\) abelian then, over \(\bar F\), the eigenvalues \(\mu \ne \pm 1\) of \(P(g)\) are simple and pair off, \(\dim E_{\pm 1}\le 2\), \(\dim E_{-1}\) is even, and \(\dim \mathfrak {so}(U)_{P(g)}=\lfloor \dim U/2\rfloor \) by induction over orthogonal splittings; the dimension descends to \(F\). Finally \(\operatorname {rank}\mathfrak {gspin}(U)=\lfloor \dim U/2\rfloor +1\): Jantzen’s lower bound [ 12 ] , and an explicit skew element \(\sum c_iJ_i\) on orthogonal planes.
For \(x\in G_{\mathrm{reg}}(F)\), \(D(x)\neq 0\).
\(N=1-\mathrm{Ad}x\) is semisimple on \(\mathfrak g\), so \(\mathfrak g=\ker N\oplus \operatorname {im}N\) with \(\ker N=\mathfrak g_x\); the characteristic polynomial of \(N\) is \(t^{\dim \mathfrak g_x}\) times a polynomial with nonzero constant term \(\pm \det (N|_{\operatorname {im}N})\), and \(\dim \mathfrak g_x=\ell \).
If \(D(x)\neq 0\) then \(x\) is semisimple with abelian centralizer in \(\mathfrak g\).
\(D(x)\neq 0\) says the generalized \(1\)-eigenspace of \(\mathrm{Ad}x\) on \(\mathfrak g\) has dimension \(\le \ell \). For the Jordan decomposition \(x=su\) in \(G(F)\), \(\mathfrak g_s\) lies in it, so \(\dim \mathfrak g_s\le \ell \); the rank clauses for \(\mathrm{GSpin}\) give \(\dim \mathfrak g_s=\ell \), \(\mathfrak g_s\) abelian, and the only unipotent element of \(G(F)\) commuting with \(s\) is \(1\), so \(u=1\) and \(x=s\) is regular semisimple.
In characteristic \(0\), \(D(x)\neq 0\) if and only if \(x\) is semisimple with abelian centralizer in \(\mathfrak g\).
The two inclusions above.
The regular semisimple set \(G_{\mathrm{reg}}(F)\) is open in \(G(F)\) and its complement has Haar measure zero.
Harish-Chandra’s theorem and the identification \(G'=G_{\mathrm{reg}}(F)\).
The Harish-Chandra character \(\theta _\pi \) of \(\pi \), the representative of \(f\mapsto \operatorname {tr}\pi (f)\) that is locally constant on \(G_{\mathrm{reg}}(F)\) (Thesis Ch. 1).
For a nilpotent orbit \(\mathcal O\subset \mathfrak g(F)\), \(\hat{j}(\mathcal O,\cdot )\) is the function representing \(f\mapsto \mu _{\mathcal O}(\hat f)\), the Fourier transform being normalized by \(b\) and \(\psi \) ( [ 5 ] §1.8, §4.1; Thesis §4.1.1). It is unique on \(\mathfrak g_{\mathrm{reg}}\).
\(\theta : G(F)\to \mathbb C\) is a quasi-character (Thesis §4.1.1, [ 5 ] §4.1) if it is conjugation invariant on \(G_{\mathrm{reg}}(F)\) and at every semisimple \(x\) there are coefficients \(c_{\theta ,\mathcal O}(x)\), \(\mathcal O\in \mathrm{Nil}(\mathfrak g_x)\), with \(\theta (xe^X)=\sum _{\mathcal O}c_{\theta ,\mathcal O}(x)\, \hat j(\mathcal O,X)\) for \(X\in \mathfrak g_x(F)\) regular and near \(0\).
For a quasi-character \(\theta \), semisimple \(x\) and \(\mathcal O\in \mathrm{Nil}_{\mathrm{reg}}(\mathfrak g_x)\), the germ \(c_{\theta ,\mathcal O}(x)\) is the coefficient of \(\hat j(\mathcal O,\cdot )\) in the local expansion of \(\theta \) at \(x\) (Thesis §4.2, [ 5 ] §4.5); it is \(0\) if \(\mathcal O\) is not regular or there is no expansion.
At semisimple \(x\), the coefficients of a local expansion \(\theta (xe^X)=\sum _{\mathcal O}c_{\mathcal O}\hat j(\mathcal O,X)\) are unique on the regular orbits \(\mathcal O\in \mathrm{Nil}_{\mathrm{reg}}(\mathfrak g_x)\) (linear independence of the \(\hat j(\mathcal O,\cdot )\) near \(0\)).
Let \(G\) be quasi-split. For each \(\mathcal O\in \mathrm{Nil}_{\mathrm{reg}}(\mathfrak g)\) there is a Whittaker datum \((U,\chi )\) such that for every irreducible tempered \(\pi \), \(c_{\theta _\pi ,\mathcal O}(1)=1\) if \(\pi \) has a \((U,\chi )\)-Whittaker model and \(0\) otherwise (Thesis Prop. 4.4.1(i)).
For a finite-dimensional \(F\)-vector space \(V\) with a nondegenerate bilinear form \(B\) there is a Haar measure \(\mu \) on \(V\) for which Fourier inversion holds: \(f(Z)=\int \hat f(Y)\psi (-B(Z,Y))\, d\mu (Y)\) with \(\hat f(Y)=\int f(X)\psi (B(X,Y))\, d\mu (X)\), for all \(f\in C_c^\infty (V)\).
Assumed from: Tate’s thesis; [ 27 ] , Ch. II §5, Ch. VII §2; [ 5 ] , §1.6; Thesis §2.4.