# Talk 3: Relative Langlands Duality 

> Abstract: If we are given a compact Lie group G acting on a space X, a powerful tool in “approximately” decomposing the G-action on functions on X is the orbit method. I will describe this method and how it sometimes refines to an exact algebraic statement which involves a “dual” group G^ and dual space X^. This is part of a joint work with David Ben-Zvi and Yiannis Sakellaridis about duality in the relative Langlands program. I will do my best to make the talk comprehensible without any familiarity with the framework of the Langlands program. 

:::{.remark}
Today: relative Langlands duality.
A toy model: let $G$ be a compact Lie group, e.g. $\SO_3$, and $X$ a compact $G\dash$space, e.g. $X=G/H$ or $X=S^2$.
The functions $L^2(X)$ o $X$ decompose as a $G\dash$representation as $L^2(X) = \bigoplus _{\pi\text{irreducible} } ^{?}$.
Goal: parameter all irreducible representations of $G$, and decide which occur.
See Kirillov's approach.
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:::{.remark}
A simpler toy model: let $G\in\Fin\Ab\Grp$ and $H\leq G$, and let $X=G/H$.
All irreps of $G$ are characters $\chi: G\to S^1$, and form a dual group $G\dual$.
Note that $G\dual \cong G$, but there is no preferred isomorphism.
Those that show are those that restrict to trivial reps on $H$, which form a subgroup.
So representations are dual to conjugacy classes here -- what is the analogous duality for Lie groups as above?
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:::{.remark}
Let $G\in\Lie\Grp$ and $\lieg=\Lie(G) = T_e(G)$, e.g. for $G=\SO_3$, differentiate $AA^t=1$ to obtain $A+A^t=0$.
Let $\lieg\dual$ be the $\RR\dash$linear dual, and note $G\actson \lieg,\lieg\dual$ by conjugation.
Note that there is a map $\exp: \lieg\to G$.
Kirillov's proposal: use $G\dash$orbits on $\lieg\dual$ as the analog of duals to conjugacy classes.
This works will for nilpotent and even non-compact semisimple groups.
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:::{.theorem title="Kirillov"}
There is an injection
\[
\Irrrep(G) &\injects \lieg\dual/G \\
\pi &\mapsto O_\pi
.\]
Pullbacks of characters of $\pi$ along $\exp$ are Fourier transforms using a unique $G\dash$invariant measure on $O_\pi$, which is normalized by a symplectic form.
Here $\exp^*(\chi)$ sends to $X\to \chi(e^X)\sqrt{\Jac(X\mapsto e^X)}$.
So the Fourier transforms give characters of irreducibles.
For noncompact groups, everything is formulated in terms of distributions instead of functions.
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:::{.example title="?"}
For $G=\SO_3$, $\lieg\cong\lieg\cong\RR^3$ parameterizing the matrices as $\mattt 0 x y {-x} 0 z {-y} {-z} 0$.
The orbits are spheres of radius $1/2, 3/2, 5/2,\cdots$ with the usual area measure induced by the Lebesgue measure on $\RR^3$, scaled by $2\pi r$.
Consider the Fourier transform of ${1\over 2\pi r} \Vol_{S^2}$:
\[
I = {1\over 2\pi r} \int_{S^2} e^{ikz} \dVol = \int_{-r}^r e^{ikz} \dz = {e^{ikz} - e^{-ikz}\over ik}
,\]
using that slices of spheres have areas $2\pi r h$.
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:::{.remark}
Which $\pi$ occur in $L^2(X)$ for $X=G/H$?
Parametrize them as orbits $O_\pi \subseteq \lieg\dual$.
If $H\leq G$ corresponds to $\lieh\leq\lieg$ and $\lieh^\perp \leq\lieg\dual$, so a guess is that $\pi$ occurs iff $O_\pi$ meets $\lieh^\perp$.
This is false in the above case, but is "asymptotically valid" -- see Duistermaat-Heckman, integrating over things of increasing size converges to the sum of multiplicities.
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:::{.remark}
The Langlands POV: there is an involution $G\mapstofrom \ldual{G}$ which e.g. matches up

- $\SO_3\mapstofrom \SU_2$
- $\U_n \mapstofrom \U_n$
- $\SO_5\mapstofrom \SP_4$.

There is a bijection between $\lieg\dual/G \mapstofrom \ldual{\lieg}/\ldual{G}$ where $\ldual{\lieg} = \Lie(\ldual{G})$, and orbits $O, O'$ match up.
Moreover $O$ occurs as some $O_\pi$ iff for any \( \lambda\in O' \) the one-parameter subgroup $t\mapsto e^{t \lambda}$ as $\RR\to\ldual{G}$ factors through $S^1 = \RR/\pi \ZZ$, with some boundary condition where $\pi/2$ is sent to a certain central involution.
Correspondingly $\pi$ matches to orbits $O_\pi\in\lieg^*/G$ match with functions $\phi: S^1\to\ldual{G}$ up to conjugacy.
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:::{.remark}
Which occur in $L^2(X)$?
For nice $X$, there is a subgroup $H\leq G$ such that $\pi$ occurs in $L^2(X)$ iff $\phi_\pi$ can be conjugated into $H$.
Then $\phi_\pi$ be conjugated in $H\dual$ iff $\phi_\pi S^1$ fixes a point on $\tilde G/\tilde H$.
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# Langlands as an arithmetic TQFT

:::{.remark}
We'll be considered with $n=4$ TQFTs.
Recall that $\TQFT_n$ assigns to $\tau_{\leq n}\Mfd$:

- $W^n \mapsto A(M)\in \CC$
- $M^{n-1} \mapsto A(M)\in\kmod$
- $(W^n, M^{n-1} \bd W^n)\mapsto (A(\bd W) \in \Vectspace, \vector v \in A(\bd W))$.
- Gluing: $L\disjoint M \mapsto \inp{L}{R} = A(L\disjoint M)$
- $M^{n-2}\mapsto \Cat$.

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:::{.remark}
Langlands takes in a reductive algebraic group e.g. $G=\GL_n$, and $R$ an arithmetic scheme, e.g. $\ZZ,\ZZ\invert{p},\ZZ\adjoint{\sqrt 2}$, curves over $\FF_q, \RR, \QQpadic, \FF_q\fps{t}$, etc.
For each pair $(G, R)$ one associates (for $R=\ZZ$, local) the vector space of automorphic functions on $G(\RR)/G(\ZZ)$, or for $R=\RR$ (global) the category $\Rep G(\RR)$.
Langlands supplies another description in terms of a dual group $\ldual{G}$.
Idea: this story resembles an equivalence of TQFTs matching $A_G$ to $B_{\ldual G}$.
If $G\actson X$ then consider the reps of $G(\RR)$ on $L^2(X)$.
There are natural pairs $G\actson X$ and $\ldual G\actson \ldual X$ which again match.
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