3 Inner forms and the main theorem
Thesis §6.1–6.4. The Langlands parameter is not modelled: the main theorem is proved for every family of packets satisfying the stability, transfer and genericity hypotheses, which is all the proof uses.
3.1 Inner forms, stability and packets
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 \(H^1(F,\mathrm{PGSpin}(W))\).
\(x\in G(F)\) 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.
\(e(G_\alpha ) = (-1)^{\mathrm{rk}_F G-\mathrm{rk}_F G_\alpha }=(-1)^{w(W)+w(W_\alpha )+w(V)+w(V_\alpha )}\) (the \(p\)-adic formula; that it equals Kottwitz’s cohomological sign is [ 14 ] ).
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 \(H^1(F,\mathrm{PGSpin}(W))\).
(STAB) \(\theta \) is constant on stable conjugacy classes of regular semisimple elements (Thesis §6.1).
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.
3.2 The main theorem
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 \(m(\pi )=m_{\mathrm{geom}}(\theta _\pi )\).
The multiplicity formula holds (Thesis Thm. 7.2.2 = Thm. 1.0.2).
By the strong induction on \(\dim G\) of Thesis §7.2, from the geometric expansion, the elliptic form of Thesis Prop. 7.4.1(ii) and elliptic isolation (Chapter 5).
Let \(G\) be quasi-split and \((\Pi _\alpha )\) as above, with finitely many inner forms. Then \(\sum _\alpha \sum _{\pi \in \Pi _\alpha } m(\pi )=1\).
By (6.7.7) the sum is the germ \(c_{\theta _1,\mathcal O_1}(1)\). By Rodier’s theorem applied to the regular orbit \(\mathcal O_1\), \(c_{\theta _1,\mathcal O_1}(1)\) is the number of members of \(\Pi _1\) with a Whittaker model for the corresponding datum, which is \(1\) by (WHITT). The multiplicity formula gives \(m(\pi ){\lt}\infty \).
(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.
The inner forms are finite in number (classification of \(p\)-adic quadratic forms). The sum of the multiplicities is \(1\), and each is a natural number, so exactly one is \(1\) and the rest vanish (Thesis §6.7). Multiplicity one is not needed for this deduction.