Depth-Zero L-Packets

Depth-Zero $L$-Packet Calculator

Enter a tame Langlands parameter using Frobenius-orbit tuples. The table identifies normalized enhancements, ordered finite compact types, and admissible-triple or induction data. Assume a characteristic-zero local field with large odd residue cardinality.

Group family:Split symplectic Sp(2R)Odd special orthogonal SO(2R+1)Unitary U(n) (unramified E/F)General linear GL(n)

Langlands parameter: (degree, [Jordan blocks], character, t, s)

Residue cardinality (enter q to keep it symbolic):

Show packet members:-- Select an option --All membersSupercuspidal membersDiscrete seriesTempered generic member

Entries per page:

Input convention

Enter tuples (d, [a₁, a₂, …], theta, t, s). Each tuple contributes

$$|\cdot|^s\rho_{[\theta],t}\otimes(S_{a_1}\oplus S_{a_2}\oplus\cdots).$$

The last two coordinates are optional: t defaults to 1 and s to 0. Repeating a Jordan block specifies its multiplicity. Thus (2, [1,1], alpha, -1) means two copies of the same Weil–Deligne summand, rather than two distinct inertial orbits. The group dimension is inferred from the standard parameter: dimensions $2R+1$, $2R$, and $n$ correspond to $\mathrm{Sp}_{2R}$, $\mathrm{SO}_{2R+1}$, and $\mathrm U_n$.

Here $S_a=\operatorname{Sym}^{a-1}(\mathbb C^2)$ is the $a$-dimensional irreducible representation of $\mathrm{SL}_2(\mathbb C)$. Each displayed block is the tensor product $\rho\otimes S_a$; its dimension is $da$, with $d=\dim\rho$.

The pair $([\theta],t)$ specifies a tame Weil representation by the convention

$$\rho_{[\theta],t} =\operatorname{Ind}_{W_{F_d}}^{W_F}\mathcal Y_{\theta,t}, \qquad \mathcal Y_{\theta,t}(\mathrm{Fr}^d)=(-1)^{d-1}t.$$

This convention specifies a Weil representation and the associated general-linear depth-zero supercuspidal. General classical compact types retain the ordered parahoric factors and their actual finite character signs; see On depth zero L-packets for classical groups, Jaime Lust and Shaun Stevens.

For unitary groups, replace $F$ by the unramified quadratic extension $E$: characters are over $\mathbb F_{q^{2d}}$, their Frobenius orbits use $q^2$, and degrees are over $W_E$.

Throughout the displayed examples, character subscripts record the extension degree over $\mathbb F_q$: for example, $\alpha_2$ is a character of $\mathbb F_{q^2}^{\times}$ and $\beta_6$ of $\mathbb F_{q^6}^{\times}$. The subscript is $d$ in Sp/SO/GL and $2d$ in U. Enter the names alpha, beta, etc.; the tuple supplies the degree. Exact @k inputs use a bracketed superscript for the exponent, such as $\theta_2^{[10]}$. The quadratic Weil-character abbreviations $1,\eta,\omega,\omega\eta$ remain as defined below.

Coordinate	Meaning
d	Degree of the actual regular Frobenius orbit, hence the Weil dimension.
[a₁, …]	Positive Jordan sizes for the $\mathrm{SL}_2$ factor; monodromy is retained.
theta	An actual named character orbit, such as alpha, beta, or alpha_1. Distinct names specify distinct orbits.
t	The determinant of Frobenius, rather than the scalar of $\mathrm{Fr}^d$. Use 1, -1, i, -i, e(k/m), or a symbolic unitary value z with inverse z*.
s	A real unramified exponent, written as a decimal or fraction, such as 3/4. Classical parameters must include its dual with exponent -s.

In Sp and odd SO, an unpaired nonquadratic name of even degree denotes a self-dual inertial orbit. Its orthogonal Weil form has t=-1, and its symplectic Weil form has t=1. These forms have the same inertia; tensoring by $\eta$ does not exchange them. In U, an unpaired name has odd degree and denotes a conjugate-self-dual orbit; its conjugate-duality sign is t for t=±1.

For a pair of distinct dual orbits, use alpha and alpha*, with inverse Frobenius values and opposite exponents. In the unitary case the star denotes the conjugate dual. The pairs must have matching Jordan sizes and multiplicities. Blocks of bad parity at exponent zero must occur with even multiplicity.

The degree-one abbreviations (1, [blocks], eta), (1, [blocks], omega), and (1, [blocks], omegaeta) specify the three nontrivial quadratic characters. Equivalently, their $(\theta,t)$ coordinates are $(1,-1)$, $(\lambda,1)$, and $(\lambda,-1)$. Here $\omega=\kappa_{-\varpi}$, in the fixed quadratic and Whittaker conventions.

Symbolic and numerical q

Leave the residue-cardinality input as q to retain it symbolically. Packet labels, finite staircase partitions, symbols, parahoric ranks, and normalized Hecke formulas are computed without evaluating $q$. Assume $F/\mathbb Q_p$, $p\ne2$, and $q>2R$ for Sp and odd SO, or $q\ge n$ for unramified U. The required bound for the particular parameter is displayed in the properties table.

Named orbits are specified mathematical data: their regularity, self-duality, and distinctness are part of the input. To check an orbit numerically instead, enter an odd prime power for q and use @k for a character exponent modulo $q^d-1$ (or $q^{2d}-1$ for U). The app then checks its exact Frobenius degree and dual orbit, identifies Frobenius-equivalent exponents, and checks the uniform residue-field bound. For example, over $\mathbb F_{29}$, @28 in degree two has a regular self-dual orbit of order 30. No numerical character-value or Gauss-sum evaluation is required to list the packet.

Whittaker normalization

The coordinates $\varepsilon$ depend on a Whittaker datum $\mathfrak w=(B,\psi_U)$: a rational Borel and a nondegenerate character of its unipotent radical, considered up to $G(F)$-conjugacy. The normalization uses the upper integral pinning and a fixed additive character $\Psi:F\to\mathbb C^\times$ that is trivial on $\varpi\mathcal O_F$ and nontrivial on $\mathcal O_F$. For Sp this is the upper datum $\mathfrak w'_1$ specified below. For unramified U use its upper quasi-split reference datum and the compatible unramified splitting characters. For odd SO the quasi-split group has a unique Whittaker-datum class. The ordered compact factors are always $B_y\perp\varpi B_z$.

More explicitly, in a symplectic basis with $\langle w_i,w_j^*\rangle=\delta_{ij}$, the upper Borel stabilizes the flag generated by $w_1,\ldots,w_R$, and

$$\psi_U(u)=\Psi\!\left( \sum_{i=1}^{R-1}\langle u w_{i+1},w_i^*\rangle +\langle u w_R^*,w_R^*\rangle\right).$$

This is $c=1$ in The local theta correspondence and the local Gan–Gross–Prasad conjecture for the symplectic-metaplectic case, Hiraku Atobe, §2.3, p.7.

Fix $\Psi$ once: its conductor alone does not specify the datum. Write $\overline\Psi$ for its residual character and keep the Gauss sum

$$g=\sum_{x\in k^\times}\lambda(x)\overline\Psi(x).$$

The finite symplectic sign is $u_x=\operatorname{sgn}(D_{\mathrm{fin}}(\tau_x)/g)$ when $b_x>0$, so a representation's finite sign must be interpreted relative to this same character and Gauss phase. This choice is retained symbolically when $q$ is symbolic.

The internal parametrization is characterized by Whittaker-normalized endoscopic transfer:

$$\operatorname{Trans}_{H_s}^{G,\mathfrak w}(S\Theta_{\phi_s}) =\sum_{\pi\in\Pi_\phi}\varepsilon_{\mathfrak w,\pi}(s)\Theta_\pi.$$

For a tempered packet on the quasi-split form, the $\mathfrak w$-generic representation has the trivial character, hence all displayed signs $+1$. Inner forms retain the relevance character and the transfer normalization from the quasi-split reference. Normalized intertwining operators use this same convention.

Changing to $\mathfrak w'$ retains the ordinary parameter and the packet, but twists its indexing by a character $\chi_{\mathfrak w,\mathfrak w'}$ of the component group:

$$\varepsilon_{\mathfrak w',\pi} =\varepsilon_{\mathfrak w,\pi}\otimes\chi_{\mathfrak w,\mathfrak w'}.$$

See Genericity and contragredience in the local Langlands correspondence, Tasho Kaletha, Theorem 3.3 and §3. Thus the all-positive row refers to the stated datum; it need not identify the same representation after changing the datum. The calculation keeps this reference fixed, and navigation preserves its signed representation data.

Reading the table

The enhancement coordinates are listed in the properties table. They are signs on distinct good-parity blocks of the tempered parameter. A multiplicity-$m$ block contributes $\varepsilon^m$ to the central relevance relation. Sp uses the product-one extension of the determinant-one component group. Odd SO and U retain $\kappa_G=\pm1$ and can display both pure inner forms. The all-positive enhancement is the tempered generic member on the quasi-split form.

The tempered generic marker and filter identify this member for tempered parameters.

Each row identifies its actual classical supercuspidal base through the ordered finite types $y,z$, in the physical convention $B_y\perp\varpi B_z$. The finite unitary factor has staircase partition $(m,m-1,\ldots,1)$ and dimension $\mathsf T(m)=m(m+1)/2$. The displayed classical cuspidal symbols are constructed with coxeter-ts; the finite character signs are retained separately from the symbols.

The signed inverse for a nonquadratic orbit, or any unitary orbit, is

$$S=(-1)^{B+D}(2B+1),\qquad m_y=\frac{|S-2D|-1}{2},\qquad m_z=\frac{|S+2D|-1}{2}.$$

Here $D$ is the signed length of the odd cuspidal chain, and $B$ is the length of the even cuspidal chain. For Sp, the four quadratic signed chains are inverted separately using the signed quadratic coordinates $h_y,h_z,j_y,j_z$. For odd SO, the reflection signs $e_0,e_1$ label the full orthogonal stabilizer factors; the mixed symplectic and odd-orthogonal sectors assume the signed quadratic-twist comparison for ordinary parameter assignments and finite character signs. The pure nonquadratic and unramified unitary calculations do not require it.

The notation $\operatorname{Disc}^{\varepsilon_0}(\mathrm{Jord}_0;\sigma)$ specifies the discrete series by its admissible triple with the displayed actual compact supercuspidal base. The superscript is replaced by the actual sign tuple on the discrete core, in its block order. These signs determine the partial sign function: adjacent ratios on surviving odd lines, and absolute signs on even lines and absent odd lines. The cuspidal-support reduction removes adjacent equal-sign blocks, removes a positive smallest even block, and compresses the remaining blocks to no-gap chains. This triple specifies the representation, including the choice when a segment induction has two discrete-series subrepresentations. Following the representation link displays the same symbol, the ordered Jordan blocks, and every construction step with its signs substituted.

$\operatorname{Ind}^{\varepsilon}$ denotes the constituent of tempered induction selected by normalized commuting intertwining operators; its superscript displays the full actual enhancement in the stated component-group block order. $\operatorname{LQ}$ denotes the Langlands quotient of the positive standard module. The zero-exponent discrete core, cuspidal enhanced parameter, reduction data, compact stabilizer, and absent-line operators can be displayed using the column visibility controls. Packets and the supercuspidal filter are computed one page at a time.

For an absent nonquadratic good Weil line with equal finite staircases $m,m$, the displayed starting operator is

$$R_{\mathfrak w}=(-1)^{m+1}\frac{2T_y-(q^{f(2m+1)}-1)}{q^{f(2m+1)}+1} =(-1)^m\frac{2T_z-(q^{f(2m+1)}-1)}{q^{f(2m+1)}+1}.$$

Here $\dim\rho=2f$; for U replace $f$ by the degree over $W_E$. For an absent ramified quadratic Sp line $\chi$ with zero minus indices on both sides, the normalized involution satisfies

$$R_{\mathfrak w}=S_y=\chi(\varpi)S_z.$$

For each side $x\in\{y,z\}$, let $N_x$ be the rank of its finite symplectic group and $\tau_x$ its actual finite cuspidal representation. Put

$$L_x=\mathrm{Sp}_{2N_x+2}(k),\qquad H_x=R_{\mathrm{GL}_1(k)\times\mathrm{Sp}_{2N_x}(k)}^{L_x} (\lambda\boxtimes\tau_x).$$

The general finite involution is defined by the Weil-character projector:

$$Z_x=\frac{1}{|L_x|}\sum_{h\in L_x} \Omega_{+,x}(h^{-1})H_x(h),\qquad S_x=\frac{\dim H_x}{\operatorname{tr}_{H_x}Z_x}Z_x-1_{H_x}.$$

Here $\Omega_{+,x}$ is the finite Weil character of $L_x$ for the quadratic space $\mathbb H^{N_x}\perp\langle1\rangle$, using the fixed residual additive character $\overline\Psi$. The two constituents of $H_x$ have equal dimension, and exactly one occurs in this Weil representation; the normalization gives eigenvalues $+1$ and $-1$. The table retains this finite-sum definition symbolically.

When $N_x=0$, that side is the principal $\mathrm{SL}_2(k)$ model. For representatives $v_r$ of $\mathbb P^1(k)$, it specializes to

$$S_x=\mathcal B_x/g,\qquad (\mathcal B_x)_{r,s}=\lambda(\det(v_r,v_s)),\qquad g=\sum_{u\in k^\times}\lambda(u)\overline\Psi(u),\qquad g^2=\lambda(-1)q.$$

Thus the equal-degree case retains the Gauss phase symbolically. The supported classical groups are split Sp, odd SO with pure inner forms, and unramified U. Even special orthogonal orientation and its separate endoscopic normalization are not implemented here.

Examples

These links fill the input and generate the table. They keep $q$ symbolic unless a number is included.

Literature examples

• Lust–Stevens: the Sp₆ packet: 16 members, including eight supercuspidals. On depth zero L-packets for classical groups, Jaime Lust and Shaun Stevens, Example 9.3, pp.32–33.

• Lust–Stevens: the two Sp₄ packets ramified character ω and ramified character ωη: four members and two supercuspidals each. On depth zero L-packets for classical groups, Jaime Lust and Shaun Stevens, Example 9.4, p.33. Their unsigned calculation identifies the union. Depth Zero Supercuspidal Representations of Classical Groups into L-packets: the Typically Almost Symmetric Case, Geo Kam-Fai Tam, §4.2, p.28 (preprint numbering), separates the packets: equal finite transvection signs give ω, opposite signs give ωη.

• Lust–Stevens: the four SO₅ assignments 1, ω, η, ω, 1, ωη, η, ωη: each has two quasi-split members and one supercuspidal. On depth zero L-packets for classical groups, Jaime Lust and Shaun Stevens, Example 9.7, p.36. The full orthogonal stabilizer is retained.

• Kaletha: a regular depth-zero SL₂ specialization: two supercuspidals, one at each vertex. Regular supercuspidal representations, Tasho Kaletha, Lemma 3.4.18, p.30, and construction (5.5), Lemma 5.3.1, §5.3, pp.79–80. This is a rank-one specialization of the construction.

• The SO₉ lift of Kaletha’s Spin₉ example: eight quasi-split supercuspidals. Supercuspidal L-packets, Tasho Kaletha, Example 4.3.1, pp.48–49. Its displayed matrices lift from PSp₈ to Sp₈ and give this SO₉ parameter at q=11. The original Spin₉ packet and its nonabelian centralizer are distinct from this lifted packet.

• The exceptional Sp₂ = SL₂ packet: the four-member biquadratic packet. On L-packets and depth for SL₂(K) and its inner form, Anne-Marie Aubert, Sergio Mendes, Roger Plymen and Maarten Solleveld, §1, p.3 (the odd-p depth-zero case), and §3.4, equation (18), p.12.

• Tam: a depth-zero U₃ packet with three degree-one orbits: four quasi-split supercuspidals. Endoscopic classification of very cuspidal representations of quasi-split unramified unitary groups, Geo Kam-Fai Tam, Theorem 6.10 and Proposition 6.11, §6.3, p.49 of the preprint. This is a depth-zero specialization with three distinct conjugate-self-dual characters.

• Tam: a depth-zero U₃ packet with one degree-three orbit: one quasi-split supercuspidal. Endoscopic classification of very cuspidal representations of quasi-split unramified unitary groups, Geo Kam-Fai Tam, Theorem 6.10 and Proposition 6.11, §6.3, p.49 of the preprint. This one-orbit specialization uses a regular character of order 1332.

Further examples

These additional examples illustrate singular parameters, induction, and the input convention.

• The singular four-member Sp₈ packet: $1\oplus\rho\otimes S_1\oplus\rho\otimes S_3$. The two supercuspidals have signs $(-,-,+)$ and $(-,+,-)$, with finite quotients $\mathrm{Sp}_6(k)\times\mathrm{Sp}_2(k)$ and the reverse. The other two members are discrete series, with signs $(+,+,+)$ and $(+,-,-)$.
• The quadratic Sp₁₀ packet: 16 members, comprising four supercuspidals and twelve other discrete series.
• An absent-line tempered packet of Sp₄: $1\oplus2\rho\otimes S_1$. Its two constituents are selected by the normalized rank-one operator, with classical base $\mathrm{Sp}_0$.
• A staircase Sp₆ packet: two supercuspidals with staircase $(2,1)$ and two discrete series induced from regular bases with staircase $(1)$.
• Odd SO₉ on both pure inner forms: $\rho\otimes S_1\oplus\rho\otimes S_3$ with symplectic $\rho$, retaining the ordered odd/even quotient and the full orthogonal stabilizer.
• Unramified U₄ on both pure inner forms: two supercuspidals with finite quotients $\mathrm U_3(k)\times\mathrm U_1(k)$ and the reverse, and two other discrete-series members.
• A non-tempered Sp₄ parameter: the positive and negative unramified twists are paired, and the row gives the positive standard module's Langlands quotient.
• A depth-zero GL₆ parameter: a singleton packet with a root-of-unity Frobenius value and real unramified exponent.
• Sp₈ with the exact character orbit over 𝔽₂₉: the app verifies the degree-two self-dual orbit instead of treating it as named input data.
• The 64-member Sp₃₀ packet: 32 compact supercuspidals and 32 discrete-series partners. The link uses symbolic large $q$.