Depth-Zero Representation Inspector

Enter a compact type, with optional discrete/tempered data and general-linear factors. Follow the construction and see how each sign selects the representation.

The result shows the packet-table symbol with its actual signs and the construction of that member. Use View depth-zero L-packet to return to its packet.

Reading the representation data

In (2,[2],alpha,1,[1]), the first 2 is the degree, the second is the Jordan size, the next 1 is the Frobenius determinant $t$ , and the final [1] is the enhancement sign. This block has dimension $2\cdot2=4$ .

Enhancement signs are $+1$ or $-1$ . A 0 in the final list means no sign coordinate. In the finite type, zero marks an absent factor or sign choice; indices and Jordan sizes can be 2 or larger.

Character subscripts give the field degree: $\alpha_2$ is a character of $\mathbb F_{q^2}^{\times}$ , and $\beta_6$ of $\mathbb F_{q^6}^{\times}$ . For U, degree $d$ gives subscript $2d$ . Enter alpha or beta without a suffix; the degree comes from the tuple. Exact input @k displays the exponent in brackets, as in $\theta_2^{[10]}$ .

For a discrete series, the steps construct the supercuspidal representation, enlarge surviving blocks, and restore removed pairs. Each step shows the actual sizes, signs, and induction. Equal-sign pairs $(+1,+1)$ and $(-1,-1)$ can give the same induction; the common sign selects different subrepresentations. The Sp₈ example displays both choices and highlights the selected one.

Compact supercuspidal input

The finite types on sides $y,z$ specify the compact induction $\sigma=\operatorname{c\!-Ind}_J^{G(F)}\Lambda$ , with lattice form $B_y\perp\varpi B_z$ . Their order and character signs matter.

For Sp, enter (a,b,u) for each side. Its quadratic finite cuspidal indices are $a=m_+$ and $b=m_-$ , with rank contribution $a(a+1)+b^2$ . When $b>0$ , $u=\pm1$ specifies the actual finite character through the normalized transvection value. When $b=0$ , enter u=0. The other finite sign is determined by

v_x=(-1)^{a_x+b_x+\sum_P\lceil m_{x,P}/2\rceil}.

For odd SO, enter (m_plus,m_minus,e0,e1) on both sides. Side $y$ is odd orthogonal; side $z$ is even orthogonal. The signs $e_0,e_1$ retain the reflection action on the full stabilizer $\mathrm S(\mathrm O(V_y)\times\mathrm O(V_z))$ . Enter zero for a missing even-orthogonal index's sign. Simultaneously twisting both full orthogonal factors by determinant describes the same restriction; the app uses the representative $e_{0,y}=+1$ .

For unramified U, the staircase types suffice; there are no extra classical quadratic-index fields.

Enter finite staircase types as (d,theta,m_y,m_z). Here theta is an actual named self-dual orbit, or an exact @k orbit with numerical q. Its degree is even in Sp/SO and odd over $W_E$ in U. Each connected finite Jordan factor has the cuspidal unitary staircase $(m,m-1,\ldots,1)$ . Omit an orbit if both indices are zero. The group size and pure inner form are inferred from these data.

Leave the discrete/tempered and general-linear fields empty for a classical supercuspidal. For GL, enter the general-linear factors directly; a single factor (d,[1],theta,t,0) specifies the depth-zero supercuspidal with tame orbit theta and Frobenius determinant t.

Other depth-zero representations

Enter optional discrete/tempered data as:

(d,[a₁,…],theta,t,[ε₁,…]).

These tuples have exponent zero. Their signs select the constituent; repeat the same sign for repeated sizes and use 0 for blocks with no sign coordinate. The app checks that reducing these blocks recovers the entered compact type, including its signs. Leave the field empty to use that type's supercuspidal parameter, or enter 0 for a trivial discrete core.

The general-linear factors use (d,[a₁,…],theta,t,s). For a classical group, require $s>0$ : the app adjoins the dual factors with exponent $-s$ and forms the Langlands quotient. Zero-exponent factors belong in the discrete/tempered field. For GL, all real exponents are allowed.

The compact type supplies the actual supercuspidal base. On the cuspidal support of discrete series for $p$ -adic quasisplit $\mathrm{Sp}(N)$ and $\mathrm{SO}(N)$ , Bin Xu, Theorem 10.1 and §10, describes the admissible-triple reduction used here.

Whittaker normalization and q

All signs use the fixed Whittaker normalization: upper integral pinning, $\Psi$ trivial on $\varpi\mathcal O_F$ and nontrivial on $\mathcal O_F$ , and the stated Gauss-sum and pure-inner-form conventions. Changing this normalization changes the sign labels, while preserving the ordinary parameter.

Leave q symbolic. The finite-type dictionary and packet labels require no numerical character evaluation. Numerical q is needed for exact @k character orbits and checks the large odd residue-field range: $q>2R$ for Sp and odd SO, and $q\ge n$ for U. The mixed quadratic Sp/SO assignments assume the signed quadratic-twist comparison, in addition to the unsigned reducibility results of Lust–Stevens.

Examples

The links below fill the representation fields and display its construction. They keep q symbolic unless a number is specified.

Lust–Stevens Example 9.3: θ₁₀ ⊠ τ: On depth zero L-packets for classical groups, Jaime Lust and Shaun Stevens, Example 9.3, pp.32–33. Ordered quotient Sp₄ × SL₂; packet size 16.
Lust–Stevens Example 9.4: equal finite transvection signs and opposite finite transvection signs: On depth zero L-packets for classical groups, Jaime Lust and Shaun Stevens, Example 9.4, p.33. The equal/opposite signs select the ω/ωη packet as in 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).
Lust–Stevens Example 9.7: a full-stabilizer SO₅ type: On depth zero L-packets for classical groups, Jaime Lust and Shaun Stevens, Example 9.7, p.36. The reflection signs select one of its four ordinary assignments.
Kaletha’s regular SL₂ specialization: Regular supercuspidal representations, Tasho Kaletha, Lemma 3.4.18 and §5.3, construction (5.5) and Lemma 5.3.1.
A hyperspecial type in the SO₉ lift of Kaletha’s Spin₉ example: Supercuspidal L-packets, Tasho Kaletha, Example 4.3.1, pp.48–49; the group here is the SO₉ lift.
An exceptional SL₂ compact type: 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, and §3.4, equation (18), p.12.
Tam’s three-orbit depth-zero U₃ specialization: a hyperspecial member of the four-member quasi-split packet. 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.
Tam’s one-orbit depth-zero U₃ specialization: the unique member of its quasi-split packet. 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.
The singular nonhyperspecial Sp₈ type: ordered staircase indices (2,1), with quotient Sp₆ × Sp₂.
A U₄ type on the other pure inner form: ordered quotient U₃ × U₁; its packet link retains the input pure inner form.
A generic discrete series of Sp₈: an admissible triple with trivial compact classical base.
A tempered constituent of Sp₄ with sign −1: select the negative eigenspace of the normalized operator.
A non-tempered Langlands quotient of Sp₄: a positive general-linear factor and the trivial Sp₀ base.
A depth-zero GL₆ Langlands quotient: a singleton packet.
The generic discrete series in the 64-member Sp₃₀ packet: all seven enhancement signs are $+1$ . Its compact base is on Sp₂₆. The sign on $\rho_{[\alpha_2],1}\otimes S_2$ selects the unique subrepresentation of $\rho_{[\alpha_2],1}|\det|^{1/2}\rtimes\sigma$ ; the other six signs come from $\sigma$ .