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Derivatives and Socles

Computes for an irreducible representation π=L(m)\pi = L(\mathfrak{m}) of GLn\mathrm{GL}_n:

  • Highest Left Derivative (GL): the highest left ∣⋅∣x|\cdot|^x-derivative of π\pi
  • Highest Right Derivative (GL): the highest right ∣⋅∣x|\cdot|^x-derivative of π\pi
  • Left Socle (GL): the socle of ∣⋅∣x×π|\cdot|^x \times \pi
  • Right Socle (GL): the socle of π×∣⋅∣x\pi \times |\cdot|^x

And for an irreducible representation of good parity π=L(m;π(ϕ,η))\pi = L(\mathfrak{m}; \pi(\phi,\eta)) of Sp2n\mathrm{Sp}_{2n} or SO2n+1\mathrm{SO}_{2n+1}:

  • Highest Derivative (Sp, SO): the highest ∣⋅∣x|\cdot|^x-derivative of π\pi
  • Highest [0,-1]-Derivative (Sp, SO): the highest Δ[0,−1]\Delta[0,-1]-derivative of π\pi
  • Highest [0,1]-Derivative (Sp, SO): the highest Z[0,1]Z[0,1]-derivative of π\pi
  • Socle (Sp, SO): the socle of ∣⋅∣x⋊π|\cdot|^x \rtimes \pi
  • [0,-1]-Socle (Sp, SO): the socle of Δ[0,−1]⋊π\Delta[0,-1] \rtimes \pi
  • [0,1]-Socle (Sp, SO): the socle of Z[0,1]⋊πZ[0,1] \rtimes \pi

Derivatives and Socles Calculator