1 General spin groups and GGP triples
This chapter follows Thesis Ch. 5. Throughout, \(F\) is a \(p\)-adic field (a finite extension of \(\mathbb {Q}_p\)) and \(\psi \) a nontrivial continuous additive character of \(F\). Quadratic spaces are nondegenerate and finite-dimensional; \(b\) is the bilinear form with \(q(v)=b(v,v)\).
1.1 Quadratic spaces and the general spin group
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.)
\(\mathfrak {gspin}(V) = \{ X \in C^+(V) : [X,v] \in V \text{ for all } v\in V\} \), a Lie subalgebra of \(C(V)\) for the commutator bracket (Thesis Prop. 5.1.3.1); \(\mathfrak {gspin}(V)=\mathfrak {so}(V)\oplus F\).
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 \(\dim V - 2w(V)\le 2\).
Proved from Tau Ceti’s Witt decomposition of a regular form ( [ 16 ] , Ch. I §4).
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).
Split off the radicals and apply Tau Ceti’s Witt cancellation for a regular summand ( [ 16 ] , Ch. I §4).
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.
Proved from Tau Ceti’s classification of regular forms over a nonarchimedean local field by rank, discriminant and Hasse invariant, and its list of the realizable invariants ( [ 16 ] , Ch. VI §2; O’Meara §63), through the fact that a \(p\)-adic field is a nonarchimedean local field.
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.
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 \(0\).)
Own derivation, with no literature input; it replaces the appeal to [ 22 ] , Thm. 8.1. If \(P(g)\) commutes with \(\bar x\), then \(gxg^{-1}=c\, x\) with \(c=\pm 1\) (the commutator lies in \(\ker P\), and the Clifford norm gives \(c^2=1\)). By base change to \(\bar K\) it suffices to show \(c=\det P(g)|_{W_{-1}}\) there (Lemma 8). The condition on \(W_{+1}\) then follows from \(\det P(g)=\det P(g)|_{W_{+1}}\cdot \det P(g)|_{W_{-1}}\): the eigenspaces \(E_\mu \) and \(E_{\mu ^{-1}}\) (\(\mu \neq \pm 1\)) are paired perfectly, so \(P(g)\) has determinant \(1\) on their sum.
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 \(\det P(g)|_{W_{-1}}=1\).
Cartan–Dieudonné on the nondegenerate spaces \(W_{+1}\) and \(W_{-1}\) writes \(P(g)|_{W_{\pm 1}}\) as products of reflections in vectors \(u_i\in W_{+1}\), \(w_j\in W_{-1}\); their number is even, and \(y=\prod u_i\prod w_j\in \mathrm{GSpin}(V)\). Since \(xv=\pm vx\) for \(v\in W_{\pm 1}\), \(xy=(-1)^{\# \{ w_j\} }yx\), and \((-1)^{\# \{ w_j\} }=\det P(g)|_{W_{-1}}\). Finally \(y^{-1}g\) commutes with \(x\) by Lemma 9.
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 \(x\).
Induction over the eigenvalues \(\mu \neq \pm 1\) of \(\bar x\). \(P(g)\) preserves \(E_\mu \); by Lemma 10 some \(\gamma \) commuting with \(x\) agrees with \(P(g)\) on \(E_\mu \) and is the identity on all \(E_\nu \), \(\nu \neq \mu ,\mu ^{-1}\). Then \(P(\gamma ^{-1}g)\) is the identity on \(E_\mu \), hence on \(E_{\mu ^{-1}}\) (the two are perfectly paired and \(P(\gamma ^{-1}g)\) is an isometry). At the end \(P(g')=1\), so \(g'\) is a scalar.
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 \(E_\nu \), \(\nu \neq \mu ,\mu ^{-1}\).
For isotropic \(u,w\) and \(1+s\, \mathrm{polar}(u,w)\neq 0\), the element \(\ell =1+s\, uw\in C(V)\) lies in \(\mathrm{GSpin}(V)\) with \(P(\ell )v=v+s\, \mathrm{polar}(w,v)\, u-t\, \mathrm{polar}(u,v)\, w\), \(t=s/(1+s\, \mathrm{polar}(u,w))\). For \(u\in E_\mu \), \(w\in E_{\mu ^{-1}}\) it commutes with \(x\) (\(xu=\mu ux\), \(xw=\mu ^{-1}wx\)), and on \(E_\mu \) it is \(v\mapsto v+s\, \mathrm{polar}(w,v)\, u\). With a basis of \(E_\mu \) and the dual basis of \(E_{\mu ^{-1}}\), these give all transvections and all elementary diagonal matrices, which generate \(\mathrm{GL}(E_\mu )\).
\(\mathrm{Nil}_{\mathrm{reg}}(\mathfrak {gspin}(V))\neq \emptyset \) iff \(\mathrm{GSpin}(V)\) is quasi-split, i.e. \(\dim V-2w(V)\le 2\).
\(\Leftarrow \): a split family \(w_{-k},\dots ,w_k\) with \(\dim V\le 2k+2\) gives the explicit regular nilpotent \(\sum _i\iota (w_{i+1})\iota (w_{-i})/2\nu \), whose centralizer has dimension \(\lfloor \dim V/2\rfloor +1\), the rank (Lemma 35). \(\Rightarrow \): for a regular nilpotent \(X\), \(N=\mathrm{ad}_V X\) has \(\dim \mathfrak {so}(V)_N=\lfloor n/2\rfloor \) (the rank of \(\mathfrak {gspin}(V)\) is \(\lfloor n/2\rfloor +1\)), so over \(\bar F\) it has minimal centralizer dimension among nilpotents (Lemma 35), hence partition \([n]\) or \([n-1,1]\) (Lemma 36); hence \(N^{m}\neq 0\) with \(m=n-1\) or \(n-2\), and the upper half of a Jordan chain of \(N\) is a totally isotropic subspace of dimension \(\lfloor (n-1)/2\rfloor \) (Thesis §5.1.4).
1.2 GGP triples
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).
\(G = \mathrm{GSpin}(W)\times \mathrm{GSpin}(V)\) (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.
\(N\subset \mathrm{GSpin}(V)\) 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).
\(H = \mathrm{GSpin}(W)\ltimes N\subset G\): 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 \(\mathbb {G}_m\).
\(\xi (h) = \psi (\lambda (h))\), 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).
\(G\) is quasi-split iff both \(W\) and \(V\) are, i.e. \(\dim U-2w(U)\le 2\) for \(U=W,V\) (Thesis §6.4).
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.
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.)
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 \(\mathcal O_{-\nu }\).
By Witt theory the relevant class is \(\nu \): if \(\dim V\) is even, \(W\) is odd and quasi-split, hence \(W=\langle a\rangle \oplus W_1\) with \(W_1\) hyperbolic and \(V=\langle a\rangle \oplus (W_1\oplus Z)\), \(W_1\oplus Z\cong \langle \nu \rangle \oplus H^{k}\); if \(\dim W=2m'\ge 4\), Witt cancellation gives \(W\cong \langle b\rangle \oplus \langle -\nu \rangle \oplus H^{m'-1}\). On the resulting split family the explicit element \(\sum _i\iota (w_{i+1})\iota (w_{-i})/2\nu \) is regular nilpotent (its centralizer has dimension the rank, Lemma 35) and lies in \(\mathcal O_{\nu }\), resp. \(\mathcal O_{-\nu }\), by a direct computation (these are the classes used in [ 17 ] , §7.1; Thesis §5.1.4).
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.
\(\mathfrak g_x=\mathfrak {gspin}(W)_x\times \mathfrak {gspin}(V)_x\), and a regular nilpotent \(X\) minimizes the centralizer dimension in each factor separately. In a factor with fixed space \(A\), \(\dim \mathfrak {gspin}_{y,Y}=\dim \mathfrak {so}(A)_{\mathrm{ad}_V Y|_A}+c+1\) where \(c\) depends only on \(\mathrm{ad}_V Y\) on \(A^\perp \), and the \(A\)-component of \(Y\) can be replaced by any skew map of \(A\). So \(\mathrm{ad}_V X|_A\) is a nilpotent of minimal centralizer dimension \(\lfloor \dim A/2\rfloor \) in \(\mathfrak {so}(A)\), and \(A\) is quasi-split as in Theorem 11 (Lemmas 35 and 36).
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.
\(\Rightarrow \) is Theorem 21. \(\Leftarrow \): the pair of explicit regular nilpotents of \(\mathfrak {gspin}(W'_x)\) and \(\mathfrak {gspin}(V'_x)\) is regular nilpotent in \(\mathfrak g_x\), because with an abelian moved part the moved contribution to \(\dim \mathfrak g_{x,Y}\) is the same for every \(Y\).