Dependencies

For \(\theta \in \mathrm{QC}(G,\chi )\), the integrand

of \(m_{\mathrm{geom},s}\) is a.e. strongly measurable on \(T(F)/A_{G,H}(F)\) (Thesis §6.5).

If \(\Pi \in R_{\mathrm{temp}}(G,\eta )\) has \(\theta _\Pi =0\) on \(G_{\mathrm{ell,reg}}(F)\), then \(\Pi \in R_{\mathrm{ind}}(G,\eta )\).

Assuming the signed count, the contribution of the class of \(T=A_{G,H}\) to the stable integral is \(c_{\theta _1,\mathcal O_1}(1)\): \(|W(T)|=1\), \(T(F)/A(F)\) is a point, the weight is \(1\) and \(D^G=\Delta =1\).

For \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\eta )\) there is a virtual tempered \(\Pi \) with central character \(\eta \) with \(\theta _\Pi =\theta _f\) on \(G_{\mathrm{ell,reg}}(F)\).

(Thesis (6.1.2).) Under the hypotheses of Theorem 71, \(c_{\theta ',\mathcal O'}(y)=c_{\theta ,\mathcal O}(x)\) for all regular \(\mathcal O,\mathcal O'\).

(Thesis (6.1.2).) Let \(G\) be quasi-split, \(\theta \) a stable quasi-character on \(G(F)\) and \(\theta '\) a stable quasi-character on \(G_\alpha (F)\) with \(\theta '(y)=\theta (x)\) for stably conjugate regular \(x,y\). Then for semisimple, strongly stably conjugate \(x\in \mathrm{GSpin}(W)(F)\), \(y\in \mathrm{GSpin}(W_\alpha )(F)\), \(c_{\theta '}(y)=c_\theta (x)\), where \(c_\theta (x)\) is the average of the regular germs.

For \(x\in \Gamma (G,H)\), \(T=T_x\) and \(s{\gt}0\), \(t\mapsto 1_{T_\natural }(t)\Delta (t)^{s-1/2}\) is integrable on \(T(F)/A_{G,H}(F)\).

There is an open, closed, invariant, \(A_G(F)\)-stable \(\Omega \subset G(F)\) containing the elliptic regular set and compact modulo conjugation and \(A_G(F)\), such that every quasi-character supported (compactly modulo conjugation) in \(\Omega \) is \(\theta _{f_0}\) on \(G_{\mathrm{reg}}(F)\) for some strongly cuspidal \(f_0\in C_c^\infty (G(F))\) supported in \(\Omega \).

Let \(f_0\in C_c^\infty (G(F))\) be strongly cuspidal. Then there is a quasi-character \(\theta \), locally integrable and locally constant on \(G_{\mathrm{reg}}(F)\), with \(\int _{G(F)}\theta f'\, dg=\int _{G(F)/A_G(F)}\int _{G(F)}f_0(x^{-1}gx)f'(g)\, dg\, dx\) for every \(f'\in C_c^\infty (G(F))\), the outer integrand being integrable.

In a nondegenerate quadratic space over a field of characteristic \(\neq 2\), every isometry of determinant \(1\) is a product \(\prod _i s_{u_i}s_{w_i}\) of pairs of reflections in anisotropic vectors.

A continuous unitary character \(\chi : A_G(F)\to \mathbb {S}^1\) trivial on \(A_{G,H}(F)\) (Thesis §7.1). It is fixed throughout the trace formula.

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).

If \(\pi \) is irreducible tempered with central character \(\eta \) on \(A_G(F)\) then \(\theta _\pi \in \mathrm{QC}(G(F),\eta )\) (Thesis Ch. 1, §4.4).

For \(\theta \in \mathrm{QC}(G,\chi )\), the integrand of \(m_{\mathrm{geom},s}\) is continuous on \(T_\natural \).

For \(\theta \in \mathrm{QC}(G,\chi )\), \(t\mapsto c_{\theta ,\mathcal O_t}(t)\) is continuous (locally constant) on \(T_\natural \).

For \(\theta \in \mathrm{QC}(G,\chi )\), \(t\mapsto D^G(t)^{1/2}c_{\theta ,\mathcal O_t}(t)\) is continuous on \(T_\natural \).

\(t\mapsto D^G(t)\) is continuous on \(T_\natural \).

\(\Pi \mapsto m(\Pi )-m_{\mathrm{geom}}(\Pi )\), extended linearly (Thesis Prop. 7.4.1(ii)). The multiplicity formula for \(\pi \) is \(\mathrm{defect}[\pi ]=0\).

Assume (HYP). The defect vanishes on \(R_{\mathrm{ind}}(G,\eta )\) (Thesis Prop. 7.4.1(i), extended linearly).

Assume (HYP). If \(\theta _\Pi =0\) on \(G_{\mathrm{ell,reg}}(F)\) then \(m(\Pi )=m_{\mathrm{geom}}(\Pi )\).

\(\Delta (x)=|\det ((P(x)-1)|_{W''_x})|_F\) (Thesis (6.5.9)).

If \(P(t')=hP(t)h^{-1}\) over \(\overline{F}\) with \(h=P(g)\circ \varphi \), then \(\Delta (t')=\Delta (t)\).

(Thesis §7.1, §10.1), for the Weil quotient measure on \(H(F)\backslash G(F)/A_G(F)\).

\(J(f+g)=J(f)+J(g)\) for \(f,g\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\) (Thesis §10.1).

For \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\), \(x\mapsto K(f,x)\) is integrable on \(H(F)\backslash G(F)/A_G(F)\) (Thesis Thm. 10.1.1(ii) with Prop. 8.4.1(iii); = Thm. 1.0.3).

Assume (HYP). If \(\theta _f=\theta _{f'}\) on \(G_{\mathrm{reg}}(F)\) then \(J(f)=J(f')\) (Thesis Prop. 12.2.3).

Assume (HYP). If \(\theta _f=\theta _{f'}\) on \(G_{\mathrm{ell,reg}}(F)\) then \(J(f)-m_{\mathrm{geom}}(\theta _f)=J(f')-m_{\mathrm{geom}}(\theta _{f'})\) (Thesis Prop. 7.4.1(ii)).

\(J^{\mathrm{Lie}}(f)=\int _{H(F)\backslash G(F)/A_G(F)}K^{\mathrm{Lie}}(f,x)\, dx\) for \(f\) on \(\mathfrak g^0(F)\) (Thesis §10.1; [ 17 ] §4, [ 5 ] §8.1).

If neither \(W\) nor \(V\) is a hyperbolic plane, \(J^{\mathrm{Lie}}(f)=J^{\mathrm{Lie}}_{\mathrm{qc}}(\theta _f)\) for every strongly cuspidal \(f\) on \(\mathfrak g^0\) (Thesis Prop. 12.4.1(ii)).

Assume (HYP), neither \(W\) nor \(V\) a hyperbolic plane. Then \(J^{\mathrm{Lie}}(f)=m^{\mathrm{Lie}}_{\mathrm{geom}}(\theta _f)\) for strongly cuspidal \(f\) on \(\mathfrak g^0\) (Thesis Thm. 12.2.2).

(Thesis Prop. 8.1.1(i), existence.) There are full isotropic flags of \(W\) and \(V\) whose stabilizer \(\bar P\) satisfies \(H\cap \bar U_{\min }=\{ 1\} \), with \(H(F)\bar P(F)\) open in \(G(F)\) and of Haar-null complement.

Assume (HYP). Every \(c\) whose semisimple part is not in \(A_G(F)\) has an invariant clopen neighbourhood that either contains no elliptic regular element or on which the geometric expansion holds.

Assume (HYP). Then \(J\) extends to some \(J_{\mathrm{qc}}\) as above (Thesis Prop. 12.2.3).

Let \(G\) be quasi-split and \(y\in \Gamma (G_\alpha ,H_\alpha )\) with \(G_{\alpha ,y}\) quasi-split. Then \(W'_{\alpha ,y}\) embeds isometrically in \(W\).

The \(\mathrm{GL}_2(F)\)-invariant measure on the regular nilpotent orbit, normalized by the self-dual measure on \(\mathfrak g/\mathfrak g_E\) ( [ 5 ] §1.6, Thesis §2.4), exists.

\(G(F)\) has a minimal parabolic subgroup \(P_0\) and a compact open subgroup \(K\) with \(G(F)=P_0(F)K\).

For a GGP datum \(\tilde d\) realizing the Levi pair \((\tilde W,\tilde V)\) of a pair of isotropic flags, there is a surjective homomorphism with central kernel from the Levi quotient \(P/U\) onto \(\prod \mathrm{GL}(Z'_{i,W})\times \prod \mathrm{GL}(Z'_{j,V})\times \tilde G(F)\) (Thesis §5.1.2, (9.3.1)–(9.3.3)).

(Thesis Prop. 12.4.1(i), Prop. 12.1.1(iv).) If neither \(W\) nor \(V\) is a hyperbolic plane, there is an invariant clopen \(\Omega \ni 1\) such that every \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\) supported in \(\Omega \) has a strongly cuspidal \(F\) on \(\mathfrak g^0\) with \(J(f)=J^{\mathrm{Lie}}(F)\) and \(m_{\mathrm{geom}}(\theta _f)=m^{\mathrm{Lie}}_{\mathrm{geom}}(\theta _F)\).

Let \(x\in \mathrm{GSpin}(W)(F)\) be semisimple with abelian moved part and \(W'_x\), \(V'_x\) quasi-split. Then some \(\mathcal O\in \mathrm{Nil}_{\mathrm{reg}}(\mathfrak g_x)\) satisfies Luo’s condition at \(x\) ( [ 17 ] §7.1).

There is an invariant clopen \(\Omega \), compact modulo conjugation and \(A_G(F)\), such that \(m_{\mathrm{geom}}(\theta )=0\) for every \(\theta \in \mathrm{QC}(G,\chi )\) vanishing on \(\Omega \cap G_{\mathrm{reg}}(F)\) (Thesis Prop. 12.1.1(i), continuity).

Let \(G\) be quasi-split. If \(\dim V\) is even and \(\ge 4\), some regular nilpotent \(X\in \mathfrak {gspin}(V)\) has image in \(\mathfrak {so}(V)\) in the orbit \(\mathcal O_\nu \) (it kills an anisotropic line \(Fd_0\) and is one Jordan block on \(U_1=d_0^\perp \cong \langle \nu \rangle \oplus H^k\)). If \(\dim W\) is even and \(\ge 4\), some regular nilpotent \(X\in \mathfrak {gspin}(W)\) has image in \(\mathcal O_{-\nu }\).

For every virtual tempered \(\Pi \) with central character \(\eta \) there is \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\eta )\) with \(\theta _f=\theta _\Pi \) on \(G_{\mathrm{ell,reg}}(F)\) (Thesis Cor. 4.8.3(i) in \(\chi \)-equivariant form). This differs from the Thesis, which uses [ 5 ] Cor. 5.7.2(iv) and Schwartz functions.

Given an isometric embedding \(W'_{\alpha ,y}\hookrightarrow W\), some \(x\in \Gamma (G,H)\) is strongly stably conjugate to \(y\).

If \(x\in \Gamma (G,H)\) and \(y\in \Gamma (G_\alpha ,H_\alpha )\) are strongly stably conjugate, then \(\mathrm{Ad}(g)\circ \psi _\alpha \) restricts to an \(F\)-rational isomorphism of topological groups \(T_x(F)\simeq T_y(F)\), the identity on scalars, \(x\mapsto y\), carrying \(T_{x,\natural }\) onto \(T_{y,\natural }\) with every \(t\) stably conjugate to its image.

For \(G\) quasi-split and \(x\in \Gamma (G,H)\) not central, some \(\beta \in H^1(F,S_x)\) has \(e(G_{q_x(\beta )})=-1\).

For \(G\) quasi-split and \(x\in \Gamma (G,H)\) not central, some \(\beta \in H^1(F,S_x)\) twists \(W\) into a form \(W_{q_x(\beta )}\not\cong W\).

If \(W''_x\) contains an \(x\)-stable anisotropic plane \(\Pi \) on which \(P(x)\) has determinant \(1\), some \(\beta \in H^1(F,S_x)\) twists \(W\) into a form \(W_{q_x(\beta )}\not\cong W\).

For \(x\in \Gamma (G,H)\) not central, some \(\beta \in H^1(F,S_x)\) twists \(W\) into a form \(W_{q_x(\beta )}\not\cong W\).

For \(x\in \Gamma (G,H)\) with \(W''_x\) of dimension \(\ge 3\) or an anisotropic plane, some \(\beta \in H^1(F,S_x)\) twists \(W\) into a form \(W_{q_x(\beta )}\not\cong W\).

Assume (HYP). For \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\eta )\) there is a virtual tempered \(\Pi _f\) with central character \(\eta \), \(\theta _{\Pi _f}=\theta _f\) on \(G_{\mathrm{ell,reg}}(F)\) and \(J(f)-m_{\mathrm{geom}}(\theta _f)=m(\Pi _f)-m_{\mathrm{geom}}(\Pi _f)\).

Every class \(\beta \in H^1(F,S_x)\) determines a unique inner form \(\alpha =q_x(\beta )\): the twist of \(W\) by a cocycle representing \(\beta \) is isometric to exactly one \(W_\alpha \).

\(\Gamma (G,H)\) meets finitely many conjugacy classes of tori (Thesis §6.5).

For semisimple \(x\in G(F)\) the set \(\mathrm{Nil}_{\mathrm{reg}}(\mathfrak g_x)\) of regular nilpotent \(G_x(F)\)-orbits is finite.

For \(x\in G_{\mathrm{reg}}(F)\), \(\dim \mathfrak g_x=\operatorname {rank}\mathfrak g=\min _{X\in \mathfrak g}\dim \mathfrak g_X\).

\(G = \mathrm{GSpin}(W)\times \mathrm{GSpin}(V)\) (Thesis §5.2), as a subgroup of \((C(W)\times C(V))^\times \). Its Lie algebra is \(\mathfrak g = \mathfrak {gspin}(W)\oplus \mathfrak {gspin}(V)\), and \(A_G\) denotes its split centre.

The index set of \(\Gamma (G,H)\): \(\mathrm{GSpin}(W)(F)\)-conjugacy classes of the tori \(T_x\), \(x\in \Gamma (G,H)\).

For every GGP triple, the geometric expansion (for every central character) and the multiplicity formula hold.

\(J(f)=m_{\mathrm{geom}}(\theta _f)\) for every \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\) (Thesis Thm. 7.2.1 = Thm. 12.2.1 = Thm. 1.0.5).

Assume (HYP). If \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\) and \(\theta _f\) vanishes near \(1\), then \(J(f)=m_{\mathrm{geom}}(\theta _f)\).

Assume (HYP). If invariant clopen sets \(\Omega _i\) are each either good (the geometric expansion holds for \(f\) supported there) or contained in the zero set of \(\theta \), and \(\theta _f\) is supported in \(\bigcup _i\Omega _i\) and in the support of \(\theta \), then \(J(f)=m_{\mathrm{geom}}(\theta _f)\) (Thesis Prop. 12.3.1, [ 5 ] §11.6).

(Thesis Prop. 12.4.1 and Thm. 12.2.2.) Assume (HYP). For every \(\chi \) there is an invariant clopen \(\Omega \ni 1\) with \(J(f)=m_{\mathrm{geom}}(\theta _f)\) for every \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\) supported in \(\Omega \).

If \(V\) is a hyperbolic plane (\(\dim W=1\), \(r=0\)), the geometric expansion holds on all of \(G(F)\).

Assume (HYP) and \(W\) a hyperbolic plane. There is an invariant clopen \(\Omega \ni 1\) on which the geometric expansion holds.

(Thesis Thm. 12.2.1 under (HYP).) If (HYP) holds for \(G\), then \(J(f)=m_{\mathrm{geom}}(\theta _f)\) for every central character \(\chi \) and every \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\).

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.

For central \(c\in A_{G,H}(F)\), \(c_{\theta _1,\mathcal O_c}(c)=c_{\theta _1,\mathcal O_1}(1)\).

For \(\theta \in \mathrm{QC}(G,\chi )\) and a torus \(T\), \(D^G(t)^{1/2}c_{\theta ,\mathcal O_t}(t)\) is bounded on \(T_\natural =\{ t\in \Gamma (G,H): T_t=T\} \).

If \(\theta \) is a stable quasi-character on \(G(F)\) and \(x\in G(F)\) is semisimple, then \(c_{\theta ,\mathcal O}(x)=c_{\theta ,\mathcal O'}(x)\) for all \(\mathcal O,\mathcal O'\in \mathrm{Nil}_{\mathrm{reg}}(\mathfrak g_x)\).

A GGP datum is a quadratic space \(W\) on \(F^m\) together with \(r\ge 0\) and \(\nu \in F^\times \). Put \(Z\) the split space of dimension \(2r+1\) with basis \(z_{-r},\dots ,z_r\), \(b(z_i,z_j)=\nu \delta _{i,-j}\), and \(V = W\oplus ^\perp Z\) (Thesis §5.2).

\(\mathfrak {gspin}(V) = \{ X \in C^+(V) : [X,v] \in V \text{ for all } v\in V\} \), a Lie subalgebra of \(C(V)\) for the commutator bracket (Thesis Prop. 5.1.3.1); \(\mathfrak {gspin}(V)=\mathfrak {so}(V)\oplus F\).

For a quadratic space \((V,q)\) with Clifford algebra \(C(V)\) and even part \(C^+(V)\), the general spin group is

(Thesis §5.1.2). Conjugation gives a surjection \(P:\mathrm{GSpin}(V)\to \mathrm{SO}(V)\) on \(\overline{F}\)-points with kernel the scalars \(\mathbb {G}_m\). (This is not the spin group of unit spinor norm.)

\(H = \mathrm{GSpin}(W)\ltimes N\subset G\): the pairs \((a, an)\) with \(a\in \mathrm{GSpin}(W)\), \(n\in N\) (Thesis §5.2). \(Z_{G,H}=A_{G,H}\) is the diagonal central \(\mathbb {G}_m\).

For every neighbourhood \(U\) of \(1\) there is an open, closed, invariant, \(A_G(F)\)-stable \(\Omega \) containing every near-central element (\(1\) in the closure of \(A_G(F)c^{G(F)}\)), each element of which is conjugate into \(U\) modulo \(A_G(F)\).

Every compact \(C\subset G(F)\) is contained in an open, closed, invariant, \(A_G(F)\)-stable \(\Omega \) that is compact modulo conjugation and \(A_G(F)\).

The regular semisimple set \(G_{\mathrm{reg}}(F)\) is open in \(G(F)\) and its complement has Haar measure zero.

If a virtual representation \(\Pi \) has the character of \(i_P^G(\sigma )\) on \(G_{\mathrm{reg}}(F)\), then \(m(\Pi )=m(i_P^G\sigma )\).

In characteristic \(0\), \(D(x)\neq 0\) if and only if \(x\) is semisimple with abelian centralizer in \(\mathfrak g\).

For \(x\in G_{\mathrm{reg}}(F)\), \(D(x)\neq 0\).

Every continuous cocycle \(\sigma \mapsto a_\sigma \in \mathrm{GL}_m(\overline{F})\) of \(\mathrm{Gal}(\overline{F}/F)\) is a coboundary: \(a_\sigma =M^{-1}\sigma (M)\).

The geometric expansion (for every central character) and the multiplicity formula hold for every GGP triple \((G',H',\xi ')\) with \(\dim G'{\lt}\dim G\) (Thesis §7.2).

An index set \(\{ \alpha \} \) with inner twists \((W_\alpha ,\varphi _\alpha )\), \(\alpha =1\) the trivial one, such that every nondegenerate form on \(F^m\) with the discriminant of \(W\) is isometric to exactly one \(W_\alpha \); so the index set is \(H^1(F,\mathrm{SO}(W))=H^1(F,H/Z_{G,H})\). \((G_\alpha ,H_\alpha ,\xi _\alpha )\) is the GGP triple of \((W_\alpha ,V_\alpha =W_\alpha \oplus Z)\). This differs from Thesis §6.2: the inner forms are indexed by \(H^1(F,\mathrm{SO}(W))\), not \(H^1(F,\mathrm{PGSpin}(W))\).

An inner twist of \(\mathrm{GSpin}(W)\): a quadratic space \(W'\) on \(F^m\) of the same dimension and discriminant with a fixed \(\overline{F}\)-isometry \(\varphi :W_{\overline{F}}\simeq W'_{\overline{F}}\) such that \(\varphi ^{-1}\circ \sigma (\varphi )\in \mathrm{SO}(W)(\overline{F})\) for all \(\sigma \). It gives \(\psi _\alpha =C(\varphi )\times C(\varphi \oplus \mathrm{id}_Z):G_{\overline{F}}\simeq G_{\alpha ,\overline{F}}\). This differs from Thesis §6.2: the inner forms are indexed by \(H^1(F,\mathrm{SO}(W))\), not \(H^1(F,\mathrm{PGSpin}(W))\).

\(x\in \mathrm{GSpin}(W)(F)\) defines an element of \(\Gamma (G,H)\) if \(x\) is semisimple, \(T_x\) is a torus and \(T_x(F)/A_{G,H}(F)\) is compact. This differs from Thesis §6.5, which requires \(T_x\) itself to be anisotropic.

A linear form \(J_{\mathrm{qc}}\) on \(\mathrm{QC}(G,\chi ^{-1})\) that depends only on values on \(G_{\mathrm{reg}}(F)\), satisfies \(J(f)=J_{\mathrm{qc}}(\theta _f)\), is continuous, and \(J_{\mathrm{qc}}-m_{\mathrm{geom}}\) is supported in the elliptic set (Thesis Prop. 12.2.3). This differs from Thesis Prop. 12.2.3, which has \(\mathrm{Supp}(J_{\mathrm{qc}})\subset G(F)_{\mathrm{ell}}\): here \(\Gamma (G,H)\) need not lie in the elliptic set.

The Haar measure of \(G(F)=\mathrm{GSpin}(W)(F)\times \mathrm{GSpin}(V)(F)\) (and of every inner form) is right invariant.

For semisimple \(x\in G(F)\), the regular semisimple elements of \(\mathfrak g_x\) form an open subset.

\(T_\natural =\{ t\in T: t\in \Gamma (G,H),\ T_t=T\} \) is open in \(T(F)\) (Thesis §6.5).

\(\theta : G(F)\to \mathbb C\) 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 \(0\).

\(G\) is quasi-split iff both \(W\) and \(V\) are, i.e. \(\dim U-2w(U)\le 2\) for \(U=W,V\) (Thesis §6.4).

For \(x\in \Gamma (G,H)\), \(P(x)\otimes \overline{F}\) is semisimple on \(W_{\overline{F}}\).

Over a field of characteristic \(0\), \(x\in \mathrm{GSpin}(V)\) is semisimple (acts semisimply on \(C(V)\)) if \(P(x)\in \mathrm{SO}(V)\) is semisimple.

(STAB) \(\theta \) is constant on stable conjugacy classes of regular semisimple elements (Thesis §6.1).

\(\pi \) 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 strongly cuspidal \(f_0\in C_c^\infty (G(F))\), the orbital pairing of its \(\chi \)-projection \(P_\chi f_0\) is represented by a quasi-character (Thesis §4.8, §7.1).

For \(\theta \in \mathrm{QC}(G,\chi )\) and \(s{\gt}0\), the integrand of \(m_{\mathrm{geom},s}(\theta )\) is integrable on each \(T(F)/A_{G,H}(F)\).

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 \(\mathfrak g_{\mathrm{reg}}\).

For \(\varepsilon \) a nonsquare and \(x\neq 0\), \(\hat j(\mathcal O_{\mathrm{reg}},x\begin{pmatrix} 0 & 1 \\ \varepsilon & 0 \end{pmatrix})=0\).

For \(X=x\, \mathrm{diag}(1,-1)\), \(x\neq 0\): \(\hat j(\mathcal O_{\mathrm{reg}},X)=2/|2x|\).

(Thesis Thm. 11.6.1, in normalization-free form.) If neither \(W\) nor \(V\) is a hyperbolic plane, \(J^{\mathrm{Lie}}(f)\) depends only on \(\theta _f|_{\mathfrak g^0_{\mathrm{reg}}}\).

There are \(d{\gt}0\) and a continuous seminorm \(\nu \) on \(\mathcal C(A_G(F)\backslash G(F),\chi ^{-1})\) with \(|K(f,x)|\le \nu (f)\, \Xi ^{H\backslash G}(x)^2\sigma _{H\backslash G}(x)^{d}\).

For every \(d{\gt}0\) there is a continuous seminorm \(\nu _d\) with \(|K(f,x)|\le \nu _d(f)\, \Xi ^{H\backslash G}(x)^2\sigma _{H\backslash G}(x)^{-d}\) for strongly cuspidal \(f\).

\(e(G_\alpha ) = (-1)^{\mathrm{rk}_F G-\mathrm{rk}_F G_\alpha }=(-1)^{w(W)+w(W_\alpha )+w(V)+w(V_\alpha )}\) (the \(p\)-adic formula; that it equals Kottwitz’s cohomological sign is [ 14 ] ).

For \(G\) quasi-split and \(x\in \Gamma (G,H)\), \(\beta \mapsto e(G_{q_x(\beta )})\) is multiplicative on \(H^1(F,S_x)\).

For \(r=0\) there is \(\epsilon {\gt}0\) with \(\int _{H_0(F)/A_{G,H}(F)}\Xi ^{G_0}(h_0)e^{\epsilon \sigma (h_0)}\, dh_0{\lt}\infty \). This differs from Thesis Lemma 8.2.1(i), which integrates over \(H_0(F)/A_{H_0}(F)\).

There is \(d{\gt}0\) with \(\int _{H(F)/A_{G,H}(F)}\Xi ^G(h)\sigma (h)^{-d}\, dh{\lt}\infty \).

For every \(\delta {\gt}0\) there is \(\epsilon {\gt}0\) with \(\int _{H(F)/A_{G,H}(F)}\Xi ^G(h)e^{\epsilon \sigma (h)}(1+|\lambda (h)|)^{-\delta }\, dh{\lt}\infty \).

(Thesis §12.5.) If \(J^{\mathrm{Lie}}(f)-m^{\mathrm{Lie}}_{\mathrm{geom}}(\theta _f)=\sum _{\mathcal O}c_{\mathcal O}c_{\theta _f,\mathcal O}(0)\) for every strongly cuspidal \(f\) on \(\mathfrak g^0\), then every \(c_{\mathcal O}\) vanishes.

For \(x\in \Gamma (G,H)\), the skew endomorphisms of \(W_{\overline{F}}\) commuting with \(P(x)\) and vanishing on \(W'_x\otimes \overline{F}\) commute.

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).

For every GGP triple and every irreducible tempered unitary representation \(\pi \) of \(G(F)\), \(m(\pi )=\dim \mathrm{Hom}_{H(F)}(\pi ^\infty ,\xi )\le 1\) (Thesis Thm. 5.5.1).

Assumed from: [ 7 ] , Thm. 1.3 (\(r=0\)), with the reduction of [ 5 ] , §6.3 and [ 9 ] , §15; directly [ 28 ] .

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.

For \(x\in \Gamma (G,H)\), \(H^1(F,S_x)\to H^1(F,SO(W)_{W'_x})\) is surjective (\(S_x\) is an anisotropic, hence fundamental, maximal torus of \(SO(W)_{W'_x}\cong SO(W''_x)\)).

Assumed from: [ 15 ] , §10, Lemma 10.2.

For \(x\in \Gamma (G,H)\), \(H^1(F,N_x)\) is finite, \(N_x=N_{\mathrm{SO}(W)}(S_x)\).

Assumed from: Borel–Serre, Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), Thm. 6.1.

For \(x\in \Gamma (G,H)\), \(H^1(F,S_x)\) is finite.

Assumed from: [ 20 ] , III §4.3 Thm. 4 (Borel–Serre); Tate–Nakayama.

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 \(0\)).

Assumed from: [ 11 ] ; [ 5 ] , §4.5; Thesis §4.2.

For an irreducible tempered \(\pi \), the distribution \(f\mapsto \operatorname {tr}\pi (f)\) on \(C_c^\infty (G(F))\) is given by a locally integrable function which is locally constant on \(G_{\mathrm{reg}}(F)\).

Assumed from: Bernstein (admissibility); [ 11 ] ; [ 5 ] , §2.1; Thesis Ch. 1.

For irreducible tempered \(\pi \), \(\theta _\pi \) is a quasi-character.

Assumed from: [ 11 ] ; [ 26 ] , §4; [ 5 ] , Prop. 4.1.1(i); Thesis Prop. 4.1.1.1(i).

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 \(G=\mathrm{GSpin}(W)\times \mathrm{GSpin}(V)\), \(\ell =(\lfloor m/2\rfloor +1)+(\lfloor (m+2r+1)/2\rfloor +1)\).

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.

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).

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)).

Assumed from: [ 19 ] ; [ 18 ] ; [ 5 ] , Prop. 4.8.1(i).

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.

(Thesis Thm. 6.4.1 = Thm. 1.0.1, \(p\)-adic case.) Let \(G\) be quasi-split and \((\Pi _\alpha )_\alpha \) tempered packets on the inner forms \(G_\alpha \), \(\alpha \in H^1(F,\mathrm{SO}(W))\), with central characters trivial on \(Z_{G,H}(F)\), satisfying (STAB), (TRANS), (WHITT). Then exactly one \(\pi \in \bigsqcup _\alpha \Pi _\alpha \) has \(m(\pi )=1\), and \(m(\pi ')=0\) for all others.

For an invariant clopen \(\Omega \), \(p\in P(F)\) and \(u\in U(F)\), \(pu\in \Omega \iff p\in \Omega \).

If \(D(x)\neq 0\) then \(x\) is semisimple with abelian centralizer in \(\mathfrak g\).

For \(s{\gt}0\),

the sum over the classes of tori of \(\Gamma (G,H)\), with the probability Haar measure and \(W(T)=N_{\mathrm{GSpin}(W)(F)}(T)/Z_{\mathrm{GSpin}(W)(F)}(T)\); and \(m_{\mathrm{geom}}(\theta )=\lim _{s\to 0^+}m_{\mathrm{geom},s}(\theta )\) (Thesis §7.2, Prop. 12.1.1). This differs from Thesis (6.5.3), which uses \(N_{\mathrm{GSpin}(W'')(F)}(T)/T(F)\) for \(W(T)\).

\(m_{\mathrm{geom}}(\theta _1+\theta _2)=m_{\mathrm{geom}}(\theta _1)+m_{\mathrm{geom}}(\theta _2)\) on \(\mathrm{QC}_{\mathrm{fin}}\) (Thesis Prop. 12.1.1(i)).

For an isomorphism of GGP triples \(e:G\to G'\) and a quasi-character \(\theta '\) of \(G'(F)\), \(m_{\mathrm{geom}}(\theta '\circ e)=m_{\mathrm{geom}}(\theta ')\) (Thesis §7.2, Prop. 12.1.1).

For tempered \(\pi _1,\dots ,\pi _k\) with central characters trivial on \(A_{G,H}(F)\), \(m_{\mathrm{geom}}(\sum _i\theta _{\pi _i})=\sum _im_{\mathrm{geom}}(\theta _{\pi _i})\) (Thesis Prop. 12.1.1, used in (6.7.2)).

\(m^{\mathrm{Lie}}_{\mathrm{geom}}(\theta )=\lim _{s\to 0^+}m^{\mathrm{Lie}}_{\mathrm{geom},s}(\theta )\) (Thesis Prop. 12.1.1(iii)).

If the signed count holds on quasi-split classes, then for \(s{\gt}0\) the stable integral equals \(c_{\theta _1,\mathcal O_1}(1)\) (Thesis §6.7 after (6.7.6)).

For a unitary representation \(\pi \) of \(G(F)\),

where \(\pi ^\infty \) is the space of smooth vectors (Thesis §5.5).

Assume (HYP). For \(\pi =i_Q^G(\sigma )\), \(Q\) proper and \(\sigma \) tempered with central character \(\eta \), \(m(\pi )=m_{\mathrm{geom}}(\theta _\pi )\).

For every irreducible tempered \(\pi \) of \(G(F)\) trivial on \(A_{G,H}(F)\): \(m(\pi ){\lt}\infty \) and \(m(\pi )=m_{\mathrm{geom}}(\theta _\pi )\).

(Thesis Prop. 7.4.1(iii), direction Thm. 7.2.1 \(\Rightarrow \) Thm. 7.2.2.) Under (HYP), if the geometric expansion holds at \(G\) for all central characters, the multiplicity formula holds at \(G\).

The multiplicity formula holds (Thesis Thm. 7.2.2 = Thm. 1.0.2).

The statement of Thesis Thm. 7.2.2 (= Thm. 1.0.2): for every GGP triple and every irreducible tempered \(\pi \) with central character trivial on \(A_{G,H}(F)\), \(m(\pi ){\lt}\infty \) and \(m(\pi )=m_{\mathrm{geom}}(\theta _\pi )\).

\(m(\pi )=m_{\mathrm{geom}}(\theta _\pi )\) (and \(m(\pi ){\lt}\infty \)) for every GGP triple and every tempered \(\pi \) trivial on \(A_{G,H}(F)\) (Thesis Thm. 7.2.2 = Thm. 1.0.2).

\(N\subset \mathrm{GSpin}(V)\) is the unipotent radical of the stabilizer \(P_V\) in \(\mathrm{GSpin}(V)\) of the isotropic flag \(\langle z_r\rangle \subset \langle z_r,z_{r-1}\rangle \subset \dots \subset \langle z_r,\dots ,z_1\rangle \) (Thesis §5.2).

\(n_\alpha (x)\) is the number of \(\mathrm{GSpin}(W_\alpha )(F)\)-conjugacy classes of \(y\in \Gamma (G_\alpha ,H_\alpha )\) strongly stably conjugate to \(x\) (Thesis §6.7).

For \(x\in \Gamma (G,H)\) and every \(\alpha \), \(n_\alpha (x)=|q_x^{-1}(\alpha )|\), \(q_x^{-1}(\alpha )\subset H^1(F,S_x)\). This differs from Thesis Prop. 6.6.1(iii), which uses \(H^1(F,T_x)\).

For semisimple \(x\in \mathrm{GSpin}(W)(F)\) whose moved part \(G''_x\) is a torus (true for every \(x\) contributing to \(\Gamma (G,H)\)): \(\mathrm{Nil}_{\mathrm{reg}}(\mathfrak g_x)\neq \emptyset \) iff \(W'_x\) and \(V'_x=W'_x\oplus Z\) are quasi-split.

For semisimple \(x\in \mathrm{GSpin}(W)(F)\) embedded diagonally in \(G\), with \(W'_x=\ker (P(x)-1)\) and \(V'_x = W'_x\oplus Z\): if \(\mathrm{Nil}_{\mathrm{reg}}(\mathfrak g_x)\ne \emptyset \) then \(W'_x\) and \(V'_x\) are quasi-split.

\(\mathrm{Nil}_{\mathrm{reg}}(\mathfrak {gspin}(V))\neq \emptyset \) iff \(\mathrm{GSpin}(V)\) is quasi-split, i.e. \(\dim V-2w(V)\le 2\).

Let \(U,U_1,U_2\) be nondegenerate. If the cocycles of \(\varphi _i\colon U_{\bar F}\cong U_{i,\bar F}\) have continuous lifts to \(\mathrm{GSpin}(U)(\bar F)\) with equal (scalar) obstructions, then \(U_1\cong U_2\) over \(F\).

The regular nilpotent orbit \(\mathcal O_x\in \mathrm{Nil}_{\mathrm{reg}}(\mathfrak g_x)\) selected by Luo’s recipe ( [ 17 ] §7.1) applied to \((G'_x,H'_x)\); a junk value if there is none (then germs vanish anyway). This differs from the choice in Thesis §5.4: it is Luo’s recipe applied to \((G'_x,H'_x)\).

For \(x\in \Gamma (G,H)\) with \(G'_x\) quasi-split, \(\mathcal O_x\in \mathrm{Nil}_{\mathrm{reg}}(\mathfrak g_x)\).

For \(G\) quasi-split, \(\mathcal O_1\in \mathrm{Nil}_{\mathrm{reg}}(\mathfrak g)\).

For \(x\in \Gamma (G,H)\), \(W''_x\) is not a line, and if it is a plane it is anisotropic.

If \(x\in \mathrm{GSpin}(W)(F)\), \(y\in \mathrm{GSpin}(W_\alpha )(F)\) are semisimple and strongly stably conjugate, then \(c_{\theta _\alpha ,\mathcal O'}(y)=e(G_\alpha )\, c_{\theta _1,\mathcal O}(x)\) for all regular \(\mathcal O,\mathcal O'\) (Thesis §6.7 after (6.7.4)).

\(\sum _{\pi \in \Pi _\alpha }m(\pi )=m_{\mathrm{geom}}(\theta _\alpha )\) for every inner form \(\alpha \).

Let \(Q\) be a nondegenerate quadratic form on \(F^m\). The isometry classes of nondegenerate forms on \(F^m\) with the same discriminant as \(Q\) are finite in number, \(n\le 2\) of them, and admit a system of representatives containing \(Q\); \(n=2\) if \(m\ge 3\), \(n=1\) if \(m\le 1\), and for \(m=2\), \(n=1\) iff \(-\det Q\) is a square.

\(i_P^G(\sigma )\): the completion of the smooth \(f:G(F)\to H_\sigma \) with \(f(pg)=\delta _P(p)^{1/2}\sigma (p)f(g)\) for \(\| f\| ^2=\int _K\| f(k)\| ^2dk\), with \(G(F)\) acting by right translation (Thesis §3.2).

Assume (HYP). Let \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\eta )\) and \(\Pi \) virtual tempered with central character \(\eta \) such that \(\theta _\Pi =\theta _f\) on \(G_{\mathrm{ell,reg}}(F)\). Then \(J(f)-m_{\mathrm{geom}}(\theta _f)=m(\Pi )-m_{\mathrm{geom}}(\Pi )\).

Assume (HYP), neither \(W\) nor \(V\) a hyperbolic plane. Then \(J^{\mathrm{Lie}}(f)-m^{\mathrm{Lie}}_{\mathrm{geom}}(\theta _f)=\sum _{\mathcal O\in \mathrm{Nil}_{\mathrm{reg}}(\mathfrak g^0)}c_{\mathcal O}\, c_{\theta _f,\mathcal O}(0)\).

Assume (HYP), and that \(J^{\mathrm{Lie}}\) factors through \(\theta _f\) by a form seeing only \(\mathfrak g^0_{\mathrm{reg}}\). Then \(J^{\mathrm{Lie}}(f)-m^{\mathrm{Lie}}_{\mathrm{geom}}(\theta _f)=\sum _{\mathcal O\in \mathrm{Nil}_{\mathrm{reg}}(\mathfrak g^0)}c_{\mathcal O}\, c_{\theta _f,\mathcal O}(0)\) for some constants \(c_{\mathcal O}\).

Let \(G\) be quasi-split and \(y\in \Gamma (G_\alpha ,H_\alpha )\) with \(G_{\alpha ,y}\) quasi-split. Then some \(x\in \Gamma (G,H)\) is strongly stably conjugate to \(y\).

If \(x\in \Gamma (G,H)\) and \(y\in \Gamma (G_\alpha ,H_\alpha )\) are strongly stably conjugate, there is a torus transfer from \(x\) to \(y\). So the \(\Gamma \)-integrals over \(T_x(F)/A(F)\) and \(T_y(F)/A(F)\) match locally.

For every compact \(C\subseteq G(F)\) with nonempty interior defining \(\Xi ^{H\backslash G}\), there is \(d{\gt}0\) with \(\int _{H(F)\backslash G(F)/A_G(F)}\Xi ^{H\backslash G}(x)^2\sigma _{H\backslash G}(x)^{-d}\, dx{\lt}\infty \).

Quasi-characters \(\theta \) on \(G(F)\) with \(\theta (ag)=\chi (a)^{-1}\theta (g)\) for \(a\in A_G(F)\), \(g\in G_{\mathrm{reg}}(F)\) (Thesis §4.1.1, §7.2).

\(R_{\mathrm{ind}}(G,\eta )\): the span of the virtual tempered representations whose character on \(G_{\mathrm{reg}}(F)\) is that of some \(i_P^G(\sigma )\), \(P\) proper, \(\sigma \) tempered with central character \(\eta \) (Thesis §7.4).

In an irreducible unitary representation of a topological group, the centre acts by a character (Thesis §3.1.2).

\(\mathcal{C}_{\mathrm{scusp}}(A_G(F)\backslash G(F),\chi ^{-1})\), \(C_c^\infty \) version: locally constant \(f\) on \(G(F)\) with \(f(ag)=\chi (a)^{-1}f(g)\) for \(a\in A_G(F)\), compactly supported modulo \(A_G(F)\), and strongly cuspidal (\(\int _{U(F)}f(pu)\, du=0\) for every proper parabolic \(P=MU\) and \(p\in P(F)\)) (Thesis Ch. 1, §4.5, §7.1).

Let \(G\) be quasi-split and \(x\in \Gamma (G,H)\) with \(G_x\) quasi-split. Then \(\sum _\alpha e(G_\alpha )n_\alpha (x)=1\) if \(x\) is central and \(0\) otherwise. This replaces the Kottwitz-sign cancellation of Thesis §6.7 after (6.7.6).

\(\sum _\alpha e(G_\alpha )\, n_\alpha (x)\) (Thesis (6.7.6)).

Under the same hypotheses,

Given germ transfer and Prop. 6.6.1(i),(ii): for \(s{\gt}0\),

(Thesis (6.7.2)–(6.7.5) before the limit).

\(x\in G(F)\) and \(y\in G_\alpha (F)\) are stably conjugate if \(y=\psi _\alpha (g x g^{-1})\) for some \(g\in G(\overline{F})\), componentwise.

If \(\bar s=\mathrm{Ad}(g)C(\varphi _\alpha )(\bar t)\) with \(g\in \mathrm{GSpin}(W_\alpha )(\overline{F})\), then \(\mathrm{diag}(t)\) and \(\mathrm{diag}(s)\) are stably conjugate in \(G\) through \(\psi _\alpha \).

Let \(x\in \mathrm{GSpin}(V)\) be semisimple, \(\bar x=P(x)\), and \(W_{\pm 1}=\ker (\bar x\mp 1)\). Then \(Z_{\mathrm{GSpin}(V)}(x)=P^{-1}(Z_{\mathrm{SO}(V)}(\bar x)^0)\); explicitly, \(g\) commutes with \(x\) iff \(P(g)\) commutes with \(\bar x\) and \(\det P(g)|_{W_{+1}} = \det P(g)|_{W_{-1}} = 1\). (Characteristic \(0\).)

Over an algebraically closed field of characteristic \(0\), let \(\bar x=P(x)\) be semisimple, and let \(P(g)\) commute with \(\bar x\) and be the identity on \(W_{+1}\oplus W_{-1}\). Then \(g\) commutes with \(x\).

Let \(\bar x=P(x)\) be semisimple over an algebraically closed field, \(\mu ^2\neq 1\), \(\mu \neq 0\). Every \(A\in \mathrm{GL}(E_\mu )\) is \(P(\gamma )|_{E_\mu }\) for some \(\gamma \in \mathrm{GSpin}(V)\) commuting with \(x\) and acting trivially on every \(E_\nu \), \(\nu \neq \mu ,\mu ^{-1}\).

Over an algebraically closed field of characteristic \(0\), let \(x\in \mathrm{GSpin}(V)\) with \(\bar x=P(x)\) semisimple, and let \(P(g)\) commute with \(\bar x\). Then \(g\) commutes with \(x\) iff \(\det P(g)|_{W_{-1}}=1\).

Let \(G\) be quasi-split. Then \(\sum _\alpha \sum _{\pi \in \Pi _\alpha }m(\pi )=c_{\theta _1,\mathcal O_1}(1)\).

Let \(G\) be quasi-split and \((\Pi _\alpha )\) as above, with finitely many inner forms. Then \(\sum _\alpha \sum _{\pi \in \Pi _\alpha } m(\pi )=1\).

Finite families \(\Pi _\alpha \) of pairwise inequivalent irreducible tempered representations of \(G_\alpha (F)\) with central characters trivial on \(Z_{G,H}(F)\), such that, with \(\theta _\alpha =\sum _{\pi \in \Pi _\alpha }\theta _\pi \) (Thesis §6.3): (STAB) each \(\theta _\alpha \) is stable; (TRANS) \(\theta _\alpha (y)=e(G_\alpha )\theta _1(x)\) for stably conjugate regular \(x\in G(F)\), \(y\in G_\alpha (F)\); (WHITT) for every Whittaker datum of \(G\) exactly one member of \(\Pi _1\) has a Whittaker model.

For \(\theta \) in the span \(\mathrm{QC}_{\mathrm{fin}}\) of the spaces \(\mathrm{QC}(G,\chi )\), the limit \(m_{\mathrm{geom}}(\theta )\) exists (Thesis Prop. 12.1.1(i)).

For \(\theta \in \mathrm{QC}(G(F),\chi )\), \(\lim _{s\to 0^+}m_{\mathrm{geom},s}(\theta )\) exists (Thesis Prop. 12.1.1(i), (12.1.4)).

For strongly cuspidal \(f\), \(\theta _f\) is the function representing the orbital pairing \(f'\mapsto \int _{G(F)/A_G(F)}\int _{G(F)}f(x^{-1}gx)f'(g)\, dg\, dx\) (Thesis §4.8, [ 5 ] §5.5); \(0\) if there is none. Convention: \(\theta _f(ag)=\chi (a)^{-1}\theta _f(g)\).

For \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\), \(\theta _f\in \mathrm{QC}(G(F),\chi ^{-1})\) (Thesis Prop. 4.8.1(ii)).

For \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\), \(\theta _f\) represents the orbital pairing of \(f\) and is a quasi-character (Thesis §4.8, Prop. 4.8.1(ii)).

For \(f\in \mathcal S_{\mathrm{scusp}}(\mathfrak g^0(F))\), the quasi-character \(\theta _f\) (Thesis §4.8; [ 5 ] §5.5).

(Thesis Thm. 10.2.1 = Thm. 1.0.4 and Prop. 7.4.1(ii), elliptic form.) Assume (HYP). For \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\eta )\), \(J(f)-m_{\mathrm{geom}}(\theta _f)=\mathrm{defect}(\Pi _f)\), where \(\Pi _f\) is a virtual tempered representation with \(\theta _{\Pi _f}=\theta _f\) on \(G_{\mathrm{ell,reg}}(F)\).

For \(x\in \Gamma (G,H)\), the group \(S_x(\overline{F})\) is commutative.

Let \(f\) be a function of the tori \(T_x\), \(x\in \Gamma (G,H)\), with \(f(T_x)=f(T_{x'})\) whenever \(x,x'\) are strongly stably conjugate to a common \(y\in \Gamma (G_\beta ,H_\beta )\), and \(f=0\) on non-quasi-split tori. Then

where \(x(T')\) is strongly stably conjugate to a generator of \(T'\) (\(f(T_{x(T')}) := 0\) if there is none).

For \(x\in \Gamma (G,H)\), \(y\in \Gamma (G_\alpha ,H_\alpha )\): an \(F\)-rational isomorphism of topological groups \(T_x(F)\simeq T_y(F)\), the identity on \(A_{G,H}\), with \(x\mapsto y\), matching the regular parts \(T_\natural \) and the strong stable classes, and preserving \(D^G\) and \(\Delta \) (the conclusion of Thesis Prop. 6.6.1(ii)). It need not preserve \(|W(T)|\).

A strongly continuous unitary representation of a topological group on a complex Hilbert space (Thesis §3.1).

For \(P\) proper and \(\sigma \) tempered, \(\theta _{i_P^G\sigma }\) vanishes on \(G_{\mathrm{ell,reg}}(F)\).

\(R_{\mathrm{temp}}(G,\eta )\otimes \mathbb C\): finite formal linear combinations of classes of irreducible tempered representations with central character \(\eta \) on \(A_G(F)\) (Thesis §3.4, §7.4), with the linear extensions \(\theta _\Pi \), \(m(\Pi )\), \(m_{\mathrm{geom}}(\Pi )\).

The measure \(m\) of Literature 30 is unique among all Borel measures.

\(D^G(x)=|\det (1-\mathrm{Ad}x)|_{\mathfrak g/\mathfrak g_x}|_F\) (Thesis §2.1), computed as the absolute value of the trailing coefficient of the characteristic polynomial of \(1-\mathrm{Ad}x\) on \(\mathfrak g\).

If \(t\in G(F)\) and \(t'\in G_\alpha (F)\) are stably conjugate then \(D^{G_\alpha }(t')=D^G(t)\).

A Whittaker datum of \(G\) (Thesis §4.4): a Borel subgroup \(B\) (stabilizer of full maximal isotropic flags of \(W\) and \(V\)) with unipotent radical \(U\), and a generic continuous unitary character \(\chi \) of \(U(F)\), i.e. its stabilizer in \(B(F)\) is \(Z_G(F)U(F)\). (The Thesis takes \(G(F)\)-conjugacy classes of such pairs.)

Over a field of characteristic \(\neq 2\), if \(U\perp X \cong U\perp Y\) then \(X\cong Y\) (all forms finite-dimensional, possibly degenerate).

A quadratic space is \(V \cong H^{w}\perp V_{\mathrm{an}}\) with \(H\) the hyperbolic plane and \(V_{\mathrm{an}}\) anisotropic, where \(w=w(V)\) is the Witt index (the maximal dimension of a totally isotropic subspace). In particular \(\dim V_{\mathrm{an}} = \dim V - 2w(V)\), so \(\mathrm{GSpin}(V)\) is quasi-split iff \(\dim V - 2w(V)\le 2\).

\(\xi (h) = \psi (\lambda (h))\), where \(\lambda (an) = \sum _{i=0}^{r-1} b(z_{-i-1}, n z_i)\) is a homomorphism \(H\to \mathbb {G}_a\) (Thesis §5.2).