Theta correspondence

The theta correspondence is a way to transfer irreducible representations between two different groups which form a dual pair inside a symplectic group. This correspondence has a global version as well as a local one; here we focus on the latter: we work with smooth (complex) representations of $p$-adic groups. Suppose $G$ and $H$ are two such groups forming a dual pair. Given an irreducible representation $\pi$ of $G$, we would like to compute its theta lift $\theta(\pi)$, which is a representation of $H$.

Dual pairs come in two flavors: type I (these involve the classical groups) and type II (general linear groups). The local theta correspondence for type II dual pairs has been explicitly described by Minguez [5]. For type I dual pairs, the work of Atobe and Gan [1] describes $\theta(\pi)$ when $\pi$ is tempered. In [2], this was extended to a description of $\theta(\pi)$ for arbitrary $\pi$.

The purpose of these programs is to demonstrate the results of [2].

Background/Notation

Groups

To simplify notation, we focus on a specific dual pair: $\text{Sp}(W) \times \text{O}(V)$.

The correspondence between other dual pairs of type I is entirely similar, and is covered in [2].

We work over a $p$-adic field $F$. Because dual pairs have symmetric roles in the correspondence, we try to keep the notation symmetric. To that end, let $\epsilon \in \{\pm 1\}$. Let

$$\begin{split} W_n &= \text{ a } (-\epsilon)\text{-symmetric vector space of even dimension }n \\ V_m &= \text{ an } \epsilon\text{-symmetric vector space of even dimension }m \end{split}$$

Let $G=G(W_n)$ (resp. $H=H(V_m)$) denote the group of isometries of $W_n$ (resp. $V_m$).

This is just a fancy way of saying the following: if $\epsilon = 1$, then $G$ is symplectic and $H$ is orthogonal. The roles of $G$ and $H$ are switched when $\epsilon = -1$.

Splitting

To set up the theta correspondence, one needs to fix a splitting of the metaplectic cover over $G \times H$. This involves characters $\chi_V$ and $\chi_W$ of $F^\times$, see [1, $\S$2.8]. We only mention this because these characters show up throughout the results; however, they do not affect the combinatorics that govern the theta lifts.

Parameters of tempered representations

The results of [1] are stated in terms of the Local Langlands Classification. Recall that the Langlands parameter of a representation $\pi$ of $G$ can be viewed as a representation $\phi$ of $W_F \times \text{SL}_2$, where $W_F$ denotes the Weil group of the field $F$. We can always decompose this representation into irreducibles, which look like $\rho \otimes S_k$; here $\rho$ is an irreducible representation of $W_F$, and $S_k$ is the unique irreducible $k$-dimensional representation of $\text{SL}_2$.

Remark

The theta correspondence only cares about $\rho \otimes S_k$ if $\rho = \chi_V$ and $k$ is odd. Therefore, you will not need to enter the entire representation $\pi$ -- just the relevant part.

The parameter $\phi$ corresponds to an $L$-packet. To identify the representation itself, more information is needed. We need a character of a certain component group attached to the parameter (see [1, $\S$3]), which amounts to putting a sign on each of the irreducible pieces of $\phi$.

Standard modules

Recall that essentially square-integrable representation of the general linear group correspond to segments $[\rho\nu^a, \rho\nu^b]$, where $\rho$ is a unitary cuspidal representation and $a, b$ are real numbers such that $b-a$ is a non-negative integer.

Let $\tau$ is a tempered representation of a classical group. A standard module is any representation of the form $\delta_k \times \dotsb \times\delta_1 \rtimes \tau,$ where

• each $\delta_i$ is an essentially square-integrable $\text{GL}$-representation, attached to $[\rho_i\nu^{a_i}, \rho_i\nu^{b_i}]$;
• $a_k+b_k \geq \dotsb \geq a_1 + b_1 > 0$.

Here we use the standard notation for parabolic induction (see [2, $\S$2.4]); furthermore, we allow $k=0$.

Any irreducible representation of $G$ is the unique irreducible quotient of a uniquely determined standard module. The results of [2] are stated in terms of the standard module of $\pi$.

Remark

Again, you will not need to enter the entire standard module of $\pi$ -- just the segments with $\rho = \chi_V$.

Relative level

We introduce a bit of notation that's slightly tricky, but highly useful: if $\pi$ is a representation of $G(W_n)$, the lift of $\pi$ to $H(V_m)$ will be denoted by $\theta_{\ell}(\pi)$, where $\ell = n + \epsilon - m$.

Note that $l$ is always odd, because $m$ and $n$ are both even.

Example 1

Suppose $\pi$ is a representation of the symplectic group, say, $\text{Sp}(14)$. We then set $\epsilon = 1$ and $n = 14$, so that $G(W_n) = \text{Sp}(14)$. Then

$$\begin{split} \theta_3(\pi) & \text{ is the lift of $\pi$ to O}(V_{12}) \\ \theta_1(\pi) & \text{ is the lift of $\pi$ to O}(V_{14}) \\ \theta_{-1}(\pi) & \text{ is the lift of $\pi$ to O}(V_{16}) \\ \theta_{-3}(\pi) & \text{ is the lift of $\pi$ to O}(V_{18}) \end{split}$$

Example 2

Suppose $\pi$ is a representation of the orthogonal space attached to a $10$-dimensional quadratic space $W_{10}$. In this case we set $\epsilon = -1$ and $n = 10$: $G(W_{10})$ is the orthogonal group. Then

$$\begin{split} \theta_5(\pi) & \text{ is the lift of $\pi$ to Sp}(4) \\ \theta_3(\pi) & \text{ is the lift of $\pi$ to Sp}(6) \\ \theta_1(\pi) & \text{ is the lift of $\pi$ to Sp}(8) \\ \theta_{-1}(\pi) & \text{ is the lift of $\pi$ to Sp}(10) \end{split}$$

If you object to this notation, here are a few reasons why you should become a fan:

• This is the notation used in [2].
• This notation allows us to forget $n$ and $m$. We already said that the theta correspondence only sees a certain part of the parameter of $\pi$. Consequently, it is not $n$ or $m$ that are relevant; it is their difference!
• By definition, $\theta_{\ell}(\theta_{-\ell}(\pi)) = \pi$, whenever $\theta_{-\ell}(\pi)\neq 0$.

Towers

If $\pi$ is a representation of the symplectic group, there are two target towers of orthogonal groups we can lift to. If $\pi$ is a representation of an orthogonal group, we can still speak of two towers; see [1, $\S$4.2] or [2, $\S$2.10].

On one of the towers, the lifts start occurring early; on the other, late. (This is a consequence of the Conservation Relation, and can be made precise). Following [1], we refer to the early (late) occurrence tower as the going-down (going-up) tower.

Remark

Identifying the going-up/going-down tower is a subtle problem; we do not discuss it here. The going-up/going-down tower for $\pi$ is inherited from the tempered part of its standard module $\tau$; the problem for tempered representation is resolved by [1, Theorem 4.1 (2)].

Theta elevator user manual

To use the Theta elevator, you need to

• Enter $\pi$
• Choose the tower (up/down) you want to lift to; see Notation→Towers.
• Choose the relative level $\ell$ you want to lift to; Notation→Relative Level

To enter $\pi$, you will need to enter information about its standard module: you will enter the segments which correspond to $\delta_i$'s, as well as the Langlands parameter of $\tau$.

Entering $\tau$

The parameter of $\tau$ breaks up into irreducibles of the form $\rho \otimes S_k$; recall that we are only interested in the terms with $\rho = \chi_V$. Therefore, you only need to enter the $k$'s that appear. Moreover, recall that each of these terms comes with a sign $\pm$. To enter the corresponding information, we use a hack: instead of entering $k$ and its sign separately, we allow $k$ to be negative. (Caveat: all isomorphic terms necessarily have the same sign.)

Enter the information about the parameter of $\tau$ by entering a comma-separated list of (possibly negative!) integers. Repetition is allowed.

Example 1

Entering [-1,3,-5] implies that the parameter of $\tau$ contains $\overset{-}{\chi_VS_1} \oplus \overset{+}{\chi_VS_3} \oplus \overset{-}{\chi_VS_5}.$ Note that the parameter might contain other irreducibles, but these are the only ones that are relevant.

Example 2

(repetition). Entering [-1,3,-5,3] implies that the parameter of $\tau$ contains $\overset{-}{\chi_VS_1} \oplus \overset{+}{\chi_VS_3} \oplus \overset{+}{\chi_VS_3} \oplus \overset{-}{\chi_VS_5}.$

Example 3

(non-example). Entering [-1,3,-5,-3] will result in an error: there are two different signs associated with $S_3$.

Entering segments

Again we only need information about the segments $[\rho\nu^{a}, \rho\nu^{b}]$ for which $\rho = \chi_V$. Enter these segments by entering a (semicolon-separated) list of ordered pairs:

$$(a_1,b_1); (a_2,b_2); \dotsc; (a_k,b_k)$$

The list doesn't have to be sorted in any specific way.

An example

We use an example to explain the input/output of the Theta Elevator.

• Tempered parameter for $\pi$: -1,3,-5
• Standard module segments for $\pi$ written in quotient notation: (1,3); (3,4); (5,5)
• Tower: Going up tower
• $\ell$: 15

The notation we are using allows us to suppress the information that does not affect the recipe for the lift; we are only entering the relevant parts of $\tau$ and the standard module. In praticular, note that we don't even need to know which group we are starting from (symplectic or orthogonal).

For example, the parameter we entered for $\tau$ is

$$\overset{-}{\chi_VS_1} \oplus \overset{+}{\chi_VS_3} \oplus \overset{-}{\chi_VS_5}$$

which looks like a parameter for $\mathrm{Sp}(8)$. However, this could just be a part of a larger parameter, possibly of an orthogonal group.

Similarly, the segments we entered to describe the standard module are only the relevant ones -- there could be other segments, but those have $\rho \neq \chi_V$. (Keep in mind that the segments we entered are those with $\rho = \chi_V$.)

Interpreting the output

The reason this partial information about $\pi$ is sufficient is that the rest of the parameter is not changed by the theta lift. Our output was:

$$\begin{split} & \theta_{-15}(Z(\left| \cdot \right|^{5}, \delta (3, 4), \delta (1, 3);\pi ( S_{1} \oplus S_{3} \oplus S_{5},(-1, 1, -1)))) \\ =& Z(\left| \cdot \right|^{7}, \delta (5, 6), \delta (3, 5), \delta (1, 4);\pi ( S_{1} \oplus S_{3} \oplus S_{5} \oplus S_{7},(1, -1, 1, -1))) \end{split}$$

which means that the lift $\theta_{-15}(\pi)$ has standard module:

$$\chi_W \left| \cdot \right|^{7} \times \chi_W \delta (5, 6) \times \chi_W \delta (3, 5) \times \chi_W \delta (1, 4) \rtimes \sigma,$$

with $\sigma$ corresponding to the parameter

$$\overset{+}{\chi_WS_1} \oplus \overset{-}{\chi_WS_3} \oplus \overset{+}{\chi_WS_5} \oplus \overset{-}{\chi_WS_5}$$

This should be interpreted as follows: where $\tau$ had $\overset{-}{\chi_VS_1} \oplus \overset{+}{\chi_VS_3} \oplus \overset{-}{\chi_VS_5},$ $\sigma$ has $\overset{+}{\chi_WS_1} \oplus \overset{-}{\chi_WS_3} \oplus \overset{+}{\chi_WS_5} \oplus \overset{-}{\chi_WS_5}$. The rest of the parameter remains unchanged, except it gets twisted by $\chi_W\chi_V^{-1}$.

The same applies to the segments in the standard module: where $\pi$ had

$$\chi_V \left| \cdot \right|^{5} \times \chi_V \delta (3, 4) \times \chi_V \delta (1, 3),$$

$\theta_{-15}(\pi)$ has

$$\chi_W \left| \cdot \right|^{7} \times \chi_W \delta (5, 6) \times \chi_W \delta (3, 5) \times \chi_W \delta (1, 4).$$

The rest of the segments remain unchaged, except for the twisting by $\chi_W\chi_V^{-1}$.

Remark

There is a small amount of information that we lose by choosing to disregard the non-important part of the parameter, and this is related to the signs on the irreducible parts of $\sigma$. If one is lifting in the symplectic → orthogonal direction, the signs are correct. If the lift is orthogonal → symplectic, the signs might need to be flipped (this depends on the target tower). Similarly, if $\ell(\tau) = -1$, then the parameter of $\sigma$ is obtained by simply adding $S_1$ to the parameter of $\tau$. The sign attached to this $S_1$ is more-or-less arbitrary in this program, because we are missing the information needed to determine it.

For the purposes of this demonstration, the effect of these minor inaccuracies is outweighed by the benefits of simplifying the parameters. For a fully precise statement of the results on lifts of tempered representations, one should consult [1].

Remarks

• The quick and dirty introduction to the theta correspondene presented here should not be taken too seriously. Kudla's notes [4] offer a great introduction (and much more). For a historical overview see [3]. A good reference for the Local Langlands Correspondence is the Appendix of [1].
• The question of which tower is going up/down is ignored in the current version of this program. I intend to address this in the next version.
• Also left for a future version are the various dual pairs of type I that are not addressed here. I hope this current version will succeed in making the results of [2] more accessible.

References

[1] H. Atobe and W.T. Gan, Local theta correspondence of tempered representations and Langlands parameters, Invent. Math., 210 (2017), pp. 341-415.
[2] P. Bakić and M. Hanzer, Theta correspondence for p-adic dual pairs of type I, J. Reine Angew. Math. 776 (2021), pp. 63-117.
[3] W.T. Gan, Theta correspondence: Recent progress and applications, in ICM Proceedings, 2014.
[4] S. Kudla, Notes on the local theta correspondence, unpublished notes (1996).
[5] A. Minguez, Correspondance de Howe explicite: paires duales de type II, Ann. Sci. Éc. Norm. Supér. (4)41 (2008), no. 5, pp. 717-741.

📄️ Theta elevator

Computes theta lifts for classical groups

📄️ Theta lifts

Lists theta lifts of a given representation