Examples of Relative Duality

Ben-Zvi—Sakellaridis—Venkatesh have formulated a conjectural duality between certain classes of Hamiltonion varieties which predict number theoretic phenomena related to automorphic period integrals and \(L\)-functions and also reflect certain boundary theories arising in physics.

In an attempt to understand their general theory, I am compiling a list of known examples of this duality. The following table lists a \(G\)-Hamiltonian variety \(M\) and its relative dual \(\check G\)-Hamiltonian variety \(\check M\). Since this is a duality, anything that appears as \(M, \check M\) in the table should also appear as \(\check M, M\), but I will sometimes list both entries to highlight the associated period integral. For now, \(G,\check G\) will both be split reductive groups.

I am not an expert on period integrals, so please email me if you would like to suggest new entries or changes to this table! And please let me know if there are errors.

\(M\) \(G\) \(\check M\) \(\check G\) Name
\(T^* H\) \(H \times H\) \(T^* (\check H\times \check H/ \check H^{\iota\mathrm{-diag}}) \), \(\iota\) is Chevalley involution \(\check H \times \check H\) Petersson inner product, Harish-Chandra Plancherel
\(T^*G/(N,\psi) \) \(G\) pt \(\check G\) Whittaker
\(T^* \mathfrak{gl}_n \) \(\mathrm{GL}_n \times \mathrm{GL}_n\) \(T^*( \mathrm{GL}_n \times \mathbb A^n )\) \( \mathrm{GL}_n \times \mathrm{GL}_n \) Tate's thesis (\(n=1\)), Godement—Jacquet
\(T^*( \mathrm{GL}_n \times \mathbb A^n )\) \( \mathrm{GL}_n \times \mathrm{GL}_n \) \(T^* \mathfrak{gl}_n \) or \(T^*(\mathrm{std}\otimes \mathrm{std})\) \(\mathrm{GL}_n \times \mathrm{GL}_n\) Rankin—Selberg (\(n=2\)), Jacquet—Piatetski-Shapiro—Shalika
\(T^* (\mathrm{GL}_n \times \mathrm{GL}_{n-1} / \mathrm{GL}_{n-1}) \) \( \mathrm{GL}_n \times \mathrm{GL}_{n-1} \) \(T^*(\mathrm{std}\otimes \mathrm{std})\) \(\mathrm{GL}_n \times \mathrm{GL}_{n-1}\) Hecke (\(n=2\)), Jacquet—Piatetski-Shapiro—Shalika
\(T^*( \mathrm{PGL}_2^{\times 3} / \mathrm{PGL}_2^{\mathrm{diag}} )\) \( \mathrm{PGL}_2^{\times 3}\) \( \mathrm{std} \otimes \mathrm{std} \otimes \mathrm{std} \) \(\mathrm{SL}_2^{\times 3}\) Jacquet, Kudla—Harris, Ichino, others
\( T^*(\mathrm{SO}_{2n} \times \mathrm{SO}_{2n+1} / \mathrm{SO}_{2n}) \) \( \mathrm{SO}_{2n} \times \mathrm{SO}_{2n+1} \) \( \mathrm{std} \otimes \mathrm{std} \) \(\mathrm{SO}_{2n}\times \mathrm{Sp}_{2n}\) Gan—Gross—Prasad
\( \mathrm{std} \otimes \mathrm{std} \) \(\mathrm{SO}_{2n} \times \mathrm{Sp}_{2n}\) \( T^*(\mathrm{SO}_{2n} \times \mathrm{SO}_{2n+1} / \mathrm{SO}_{2n}) \) \( \mathrm{SO}_{2n} \times \mathrm{SO}_{2n+1} \) Theta correspondence
\( T^*(\mathrm{GL}_{2n} / (\mathrm{GL}_n \times \mathrm{GL}_n )) \) \(\mathrm{GL}_{2n} \) \( T^*((\mathrm{GL}_{2n} / \mathrm{Sp}_{2n}) \times \mathrm{std}) \) \( \mathrm{GL}_{2n} \) Friedberg—Jacquet (Linear period)
\( T^*(\mathrm{GL}_{2n} / \mathrm{GL}_n^{\mathrm{diag}} (U,\psi) ) \), \(U=\mathfrak{gl}_n\); Shalika model \(\mathrm{GL}_{2n} \) \( T^*(\mathrm{GL}_{2n} / \mathrm{Sp}_{2n}) \)
not tempered
\( \mathrm{GL}_{2n} \) Jacquet—Shalika
\(T^*( G/U )\) \( G \times T \) \(T^*(\check G/ \check U) \) \(\check G\times \check T \) Eisenstein series
\(T^*( \widetilde{G/U} )\), \(\widetilde{G/U}\) is "stacky" resolution of \(\overline{G/U}\) \( G \times T \) \(T^*( \check G/ \check T) \) \(\check G \times \check T\) Compactified/normalized Eisenstein series
\(T^*( \overline{G/U} )\) \( G\times T \) \(T^*( \check G/ [\check U, \check U]) \) \(\check G \times \check T\)
\( T^* \widetilde{M_\rho} \)
\(M_\rho\) is L-monoid
\( G\times G \)
\(G\) with "det" map
\( T^*(\check G \times V_\rho ) \)
\(\rho\) is irreducible \(\check G\)-representation
\(\check G \times \check G\) Braverman—Kazhdan
\( T^*( \mathrm{SO}_{2n+1}/ \mathrm{SO}_{2n}) \) \( \mathrm{SO}_{2n+1} \) \( \mathrm{Sp}_{2n} \times^{\mathrm{SL}_2} (\mathrm{std}_2[\frac 3 2 - n] \oplus \mathrm{std}_2[\frac 1 2] \oplus \mathrm{triv}[-1,-3,\dotsc,3-2n]) \) \( \mathrm{Sp}_{2n} \) Sakellaridis