Examples in (generalized) relative Langlands duality
Ben-Zvi, Sakellaridis, and Venkatesh have recently proposed a matching between objects in the Langlands program, namely that the relationship between \(G\) and its Langlands dual \(\check{G}\) should extend to a duality between Hamiltonian \(G\)-spaces \(M\) and Hamiltonian \(\check{G}\)-spaces \(\check{M}\). This goes under the name of “relative Langlands duality”. By now, others have compiled comprehensive lists (e.g., here and here) of examples of such dualities.
In two recent articles (here and here), I proposed an extension of this duality to exotic coefficients on one side, which amounts to a variant of the notion of a Hamiltonian space on the dual side. Namely, if \(X\) is a \(G\)-space, then \(M = T^\ast X\) is a Hamiltonian \(G\)-space, and the “usual” picture of relative Langlands proposes that there is a dual Hamiltonian \(\check{G}\)-space \(\check{M}\) which controls the harmonic analysis of \(X\), in the sense that (say) \(\mathrm{Shv}(X_F/G_O; \mathbf{C})\) is equivalent to \(\mathrm{QCoh}(\check{M}/\check{G})\). If one replaces \(\mathbf{C}\) by complex K-theory \(\mathrm{KU}\) or by an elliptic cohomology theory \(k\) with associated elliptic curve \(E\), then \(\check{M}\) must accordingly be replaced by a quasi-Hamiltonian or elliptic Hamiltonian \(\check{G}\)-space, respectively. The table below is a list of examples of these exotic Langlands duals. (Note that I am assuming throughout that my starting data is \(X\), not \(M\).)
For simplicity, I will ignore the case of elliptic cohomology, and focus instead on the generalized cohomology theory \(\mathrm{ku}\) called “connective complex K-theory”: the Langlands dual side will involve a graded scheme \(\check{M}_\beta\) over \(\mathbf{A}^1/\mathbf{G}_m\) with coordinate \(\beta\), whose special fiber over \(\beta = 0\) produces the “usual” Langlands dual Hamiltonian \(\check{G}\)-space, and whose generic fiber over \(\beta = 1\) produces the quasi-Hamiltonian/\(\mathrm{KU}\)-theoretic dual. The phrase “for ordinary cohomology” below means that I have not (yet) worked out the \(\mathrm{ku}\)-theoretic dual to \(X\), but that it should be interesting to do so. There is a lot more information about these examples which I did not include, like power operations (namely, Steenrod and Adams operations); for this, see Section 9 of this article.
In the table below, \(T\) denotes a maximal torus, \(B\) a Borel subgroup, and \(N\) its unipotent radical; I will assume any unspecified reductive group is connected and simply-laced. Also, \(K_\beta\) for a group scheme \(K\) will denote the deformation to the normal cone of the identity \(1\in K\), so that the special fiber of \(K_\beta\) over \(\beta = 0\) is the Lie algebra \(\mathfrak{k}\), and the generic fiber of \(K_\beta\) over \(\beta = 1\) is \(K\) itself. For example, \(\mathrm{SL}_{n,\beta}\) is the group scheme of \(n\times n\)-matrices \(A\) such that \(\frac{\det(I + \beta A) - 1}{\beta} = 0\). If \(V\) and \(W\) are vector spaces, let \(\mathcal{B}_\beta(V,W)\) denote the space of pairs \((x,y) \in T^\ast(V \otimes W^\ast)\) such that \(I + \beta \langle x,y\rangle\) is invertible; when \(\beta = 1\), this is a multiplicative quiver variety.
| \(G\) | \(X\) | \(\check{G}\) | \(\check{M}_\beta\) | Moment map \(\check{M}_\beta \to {G}_\beta\) |
|---|---|---|---|---|
| \(H \times H\) | \(H\) | \(\check{H} \times \check{H}\) | \({H}_\beta\) | Chevalley-twisted diagonal |
| \(G\) | \(G/_\psi N\) | \(\check{G}\) | \(\ast\) | Identity |
| \(\mathbf{G}_m\) | \(\mathbf{A}^1\) | \(\mathbf{G}_m\) | \(\mathcal{B}_\beta(\mathbf{A}^1, \mathbf{A}^1)\) | \((u,v)\) is sent to \(1 + \beta \langle u,v\rangle\) |
| \(H \times T\) | \(H/N\) | \(\check{H} \times \check{T}\) | \({\check{H} \times^{\check{N}} B_\beta}\) | Adjoint action |
| \(H \times T\) | \(H\) | \(\check{H} \times \check{T}\) | Affine closure of \({\check{H} \times^{\check{N}} B_\beta}\) | Adjoint action |
| \(\mathrm{PGL}_2\) | \(\mathrm{PGL}_2/\mathbf{G}_m\) | \(\mathrm{SL}_2\) | \(\mathcal{B}_\beta(\mathbf{A}^1, \mathbf{A}^2)\) | \((u,v)\) is sent to \(I + \beta u \otimes v\) |
| \(\mathrm{GL}_n \times \mathrm{GL}_n\) | \(\mathrm{GL}_n \times \mathbf{A}^n\) | \(\mathrm{GL}_n \times \mathrm{GL}_n\) | \(\mathcal{B}_\beta(\mathbf{A}^n, \mathbf{A}^n)\) | \((u,v)\) is sent to \((I + \beta uv, I + \beta vu)\) |
| \(\mathrm{GL}_n \times \mathrm{GL}_{n-1}\) | \(\mathrm{GL}_n\) | \(\mathrm{GL}_n \times \mathrm{GL}_{n-1}\) | \(\mathcal{B}_\beta(\mathbf{A}^n, \mathbf{A}^{n-1})\) | \((u,v)\) is sent to \((I + \beta uv, I + \beta vu)\) |
| \(\mathrm{GL}_{2n}\) | \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\) | \(\mathrm{GL}_{2n}\) | \(\mathrm{GL}_{2n} \times^{\mathrm{GL}_n} \mathfrak{gl}_n\), where \(\mathrm{GL}_n\) is embedded into \(\mathrm{GL}_{2n}\) by \(\mathrm{diag}(g,g)\) | \(g \mapsto \begin{pmatrix} I + \beta^2 g & \beta I \\ \beta g & I \end{pmatrix}\) |
| \(\mathrm{PSO}_{2n} \times \mathrm{SO}_{2n+1}\) | \(\mathrm{SO}_{2n+1}\) | \(\mathrm{Spin}_{2n} \times \mathrm{Sp}_{2n}\) | \(u\in \mathrm{Hom}(\mathbf{A}^{2n}, \mathbf{A}^{2n})\) such that \(I + \beta u u^\ast\) is invertible | \((u,v)\) is sent to \((I + \beta u u^\ast, I + \beta u^\ast u)\) |
| \(\mathrm{PSO}_{2n} \times \mathrm{SO}_{2n-1}\) | \(\mathrm{PSO}_{2n}\) | \(\mathrm{Spin}_{2n} \times \mathrm{Sp}_{2n-2}\) | \(u\in \mathrm{Hom}(\mathbf{A}^{2n-2}, \mathbf{A}^{2n})\) such that \(I + \beta u u^\ast\) is invertible | \((u,v)\) is sent to \((I + \beta u u^\ast, I + \beta u^\ast u)\) |
| \(\mathrm{SO}_3^{\times 3}\) | \(\mathrm{SO}_3^{\times 3}/\mathrm{SO}_3^\mathrm{diag}\) | \(\mathrm{SL}_2^{\times 3}\) | \(u\in \mathrm{Hom}(\mathbf{A}^2, (\mathbf{A}^2)^{\otimes 2})\) such that \(I + \beta u u^\ast\) is invertible | Bhargava cubes (see here for elaboration) |
| (Nonsplit) \(G(\!(t)\!)\) for \(G = \mathrm{PGL}_2\) | \(G(\!(t)\!)/G(\!(t^3)\!)\) | \(\mathrm{SL}_2\) | Orbit closure in \(\mathrm{Sym}^3(\mathbf{A}^2)\) of the parametric family \(a\mapsto ax^3+3xy^2-\beta y^3\) | Quadratic resolvent |
| (Nonsplit) \(G(\!(t)\!)\) for \(G = \mathrm{PGL}_2\) | \(G(\!(t)\!)/G(\!(t^n)\!)\), \(n>1\) odd | \(\mathrm{SL}_2\) | Affine cone on secant variety of rational normal curve \(\mathbf{P}^1 \hookrightarrow \mathbf{P}^n\) (for ordinary cohomology) | Moment map on \(\mathrm{Sym}^n(\mathbf{A}^2)\) |
Some comments: I expect the Whittaker line to be true, but have not worked it out. In the line involving \(H \times T\) acting on \(H\), \(\mathrm{QCoh}(\check{M}_\beta/\check{G})\) is only equivalent to the full subcategory of the corresponding geometric sheaf category which is generated under the spherical Hecke action by the \(\delta\)-sheaf of \(X_O \subseteq X_F\). Similarly for the nonsplit examples above. Some of these \(\mathrm{ku}\)-theoretic analogues, like the Gan-Gross-Prasad period (concerning \(\mathrm{SO}_{2n} \times \mathrm{SO}_{2n+1}\) acting on \(\mathrm{SO}_{2n+1}\)), have not yet been written up in any public document; but they can be deduced using techniques similar to those used in some of my work (e.g., in this article). The “triple product period” is a special case of the Gan-Gross-Prasad period, corresponding to \(n=2\). The above table isn’t comprehensive; there are other examples, like some rank one spherical varieties, which can be found in this article. Also, the table clearly mostly contains only homogeneous \(G\)-spaces, and is missing Whittaker-induced examples; this is not a mathematical problem, but is rather due to my laziness.
For the examples in the above table, I have verified the \(\mathrm{ku}\)-theoretic analogue of the first conjecture of this paper (which corresponds to a part of Conjecture 3.5.11 here); it roughly says that the set of \(B\)-orbit closures in \(X\) is in bijection with the set of irreducible components of \(\check{M}_\beta \times_{G_\beta} N_\beta\). This is an important sanity check for any candidate dual.