Degenerate Principal Series and the Geometry of Null Fiber

Kyo Nishiyama (Kyoto University)

NORTh 2007 2007/3/6 – 3/9

Ex 1. Hasse diagram of the closure relation top line = irreducible components

n = 4 ≤ m

\xymatrix{𝔑_{0,4} \ar@{-->}[dr] &  & 𝔑_{1,3} \ar[dl] \ar@{-->}[dr] &  & 𝔑_{2,2} \ar[dl] \ar@{-->}[dr] &  & 𝔑_{3,1} \ar[dl] \ar@{-->}[dr] &  & 𝔑_{4,0} \ar[dl] \\  & 𝔑_{0,3} \ar@{-->}[dr] &  & 𝔑_{1,2} \ar[dl] \ar@{-->}[dr] &  & 𝔑_{2,1} \ar[dl] \ar@{-->}[dr] &  & 𝔑_{3,0} \ar[dl] \\  &  & 𝔑_{0,2} \ar@{-->}[dr] &  & 𝔑_{1,1} \ar[dl] \ar@{-->}[dr] &  & 𝔑_{2,0} \ar[dl] \\  &  &  & 𝔑_{0,1} \ar@{-->}[dr] &  & 𝔑_{1,0} \ar[dl] \\  &  &  &  & 𝔑_{0,0}}

n = 4, m = 3

\xymatrix{ & 𝔑_{1,3} \ar[dl] \ar@{-->}[dr] &  & 𝔑_{2,2} \ar[dl] \ar@{-->}[dr] &  & 𝔑_{3,1} \ar[dl] \ar@{-->}[dr] \\ 𝔑_{0,3} \ar@{-->}[dr] &  & 𝔑_{1,2} \ar[dl] \ar@{-->}[dr] &  & 𝔑_{2,1} \ar[dl] \ar@{-->}[dr] &  & 𝔑_{3,0} \ar[dl] \\  & 𝔑_{0,2} \ar@{-->}[dr] &  & 𝔑_{1,1} \ar[dl] \ar@{-->}[dr] &  & 𝔑_{2,0} \ar[dl] \\  &  & 𝔑_{0,1} \ar@{-->}[dr] &  & 𝔑_{1,0} \ar[dl] \\  &  &  & 𝔑_{0,0}}

n = 4, m = 2 (stable range)

\xymatrix{ &  & 𝔑_{2,2} \ar[dl] \ar@{-->}[dr] \\  & 𝔑_{1,2} \ar[dl] \ar@{-->}[dr] &  & 𝔑_{2,1} \ar[dl] \ar@{-->}[dr] \\ 𝔑_{0,2} \ar@{-->}[dr] &  & 𝔑_{1,1} \ar[dl] \ar@{-->}[dr] &  & 𝔑_{2,0} \ar[dl] \\  & 𝔑_{0,1} \ar@{-->}[dr] &  & 𝔑_{1,0} \ar[dl] \\  &  & 𝔑_{0,0}}

n = 6 : p = q = 6 Hasse diagram of submodules of I(6)

\xymatrix{ &  &  & • \ar[dl] \ar[dr] \\  &  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] \\  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] \\ • &  & • &  & • &  & •}

∘ : non-unitary , • : unitary

Hasse diagram of associated varieties:

\xymatrix{ &  &  & 𝕆_{1,1} \ar[dl] \ar@{-->}[dr] \\  &  & 𝕆_{1,3} \ar[dl] \ar@{-->}[dr] &  & 𝕆_{3,1} \ar[dl] \ar@{-->}[dr] \\  & 𝕆_{1,5} \ar[dl] \ar@{-->}[dr] &  & 𝕆_{3,3} \ar[dl] \ar@{-->}[dr] &  & 𝕆_{5,1} \ar[dl] \ar@{-->}[dr] \\ 𝕆_{0,6} &  & 𝕆_{2,4} &  & 𝕆_{4,2} &  & 𝕆_{6,0}}

n = 6 : p = q = 7 Hasse diagram of submodules of I(7)

\xymatrix{ &  &  & • \ar[dl] \ar[dr] \\  &  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] \\  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] \\ ∘ \ar[dr] &  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \\  & • &  & • &  & •}

• : unitary , ∘ : non-unitary

Hasse diagram of associated varieties:

\xymatrix{ &  &  & 𝕆_{0,0} \ar[dl] \ar@{-->}[dr] \\  &  & 𝕆_{0,2} \ar[dl] \ar@{-->}[dr] &  & 𝕆_{2,0} \ar[dl] \ar@{-->}[dr] \\  & 𝕆_{0,4} \ar[dl] \ar@{-->}[dr] &  & 𝕆_{2,2} \ar[dl] \ar@{-->}[dr] &  & 𝕆_{4,0} \ar[dl] \ar@{-->}[dr] \\ 𝕆_{0,6} \ar@{-->}[dr] &  & 𝕆_{2,4} \ar[dl] \ar@{-->}[dr] &  & 𝕆_{4,2} \ar[dl] \ar@{-->}[dr] &  & 𝕆_{6,0} \ar[dl] \\  & 𝕆_{1,5} &  & 𝕆_{3,3} &  & 𝕆_{5,3}}

n = 6 : p = 7, q = 6 Hasse diagram of submodules of I(13/2)

\xymatrix{ &  &  & • \ar[dl] \ar[dr] \\  &  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] \\  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] \\ ∘ &  & ∘ &  & ∘ &  & •}

∘ : non-unitary , • : unitary

Hasse diagram of associated varieties:

\xymatrix{ &  &  & 𝕆_{1,0} \ar[dl] \ar[dr] \\  &  & 𝕆_{1,2} \ar[dl] \ar[dr] &  & 𝕆_{3,0} \ar[dl] \ar[dr] \\  & 𝕆_{1,4} \ar[dl] \ar[dr] &  & 𝕆_{3,2} \ar[dl] \ar[dr] &  & 𝕆_{5,0} \ar[dl] \ar[dr] \\ 𝕆_{0,6} ∪ 𝕆_{1,5} &  & 𝕆_{2,4} ∪ 𝕆_{3,3} &  & 𝕆_{4,2} ∪ 𝕆_{5,1} &  & 𝕆_{6,0}}

n = 6 : p = 8, q = 7 Hasse diagram of submodules of I(15/2)

\xymatrix{ &  &  & • \ar[dl] \ar[dr] \\  &  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] \\  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] \\ • &  & ∘ &  & ∘ &  & ∘}

∘ : non-unitary , • : unitary

Hasse diagram of associated varieties:

\xymatrix{ &  &  & 𝕆_{0,1} \ar[dl] \ar[dr] \\  &  & 𝕆_{0,3} \ar[dl] \ar[dr] &  & 𝕆_{2,1} \ar[dl] \ar[dr] \\  & 𝕆_{0,5} \ar[dl] \ar[dr] &  & 𝕆_{2,3} \ar[dl] \ar[dr] &  & 𝕆_{4,1} \ar[dl] \ar[dr] \\ 𝕆_{0,6} &  & 𝕆_{1,5} ∪ 𝕆_{2,4} &  & 𝕆_{3,3} ∪ 𝕆_{4,2} &  & 𝕆_{5,1} ∪ 𝕆_{6,0}}

n = 7 : p = q = m ≥ 7 is odd Hasse diagram of submodules of I(m)

\xymatrix{ &  &  & ⊗ \ar[dl] \ar[dr] \\  &  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] \\  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] \\ ∘ &  & ∘ &  & ∘ &  & ∘}

∘ : non-unitary , ⊗ : unitary iff m = n

Hasse diagram of associated varieties:

\xymatrix{ &  &  & 𝕆_{1,1} \ar[dl] \ar[dr] \\  &  & 𝕆_{1,3} \ar[dl] \ar[dr] &  & 𝕆_{3,1} \ar[dl] \ar[dr] \\  & 𝕆_{1,5} \ar[dl] \ar[dr] &  & 𝕆_{3,3} \ar[dl] \ar[dr] &  & 𝕆_{5,1} \ar[dl] \ar[dr] \\ 𝕆_{0,7} ∪ 𝕆_{1,6} &  & 𝕆_{2,5} ∪ 𝕆_{3,4} &  & 𝕆_{4,3} ∪ 𝕆_{5,2} &  & 𝕆_{6,1} ∪ 𝕆_{7,0}}

n = 7 : p = q = m ≥ 8 is even Hasse diagram of submodules of I(m)

\xymatrix{ &  &  &  & ⊗ \ar[dl] \ar[dr] \\  &  &  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] \\  &  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] \\  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] &  & ∘ \ar[dl] \ar[dr] \\ • &  & ∘ &  & ∘ &  & ∘ &  & •}

• : unitary , ∘ : non-unitary , ⊗ : finite dimensional, unitary iff m = n + 1

Hasse diagram of associated varieties:

\xymatrix{ &  &  &  & 𝕆_{0,0} \ar[dl] \ar[dr] \\  &  &  & 𝕆_{0,2} \ar[dl] \ar[dr] &  & 𝕆_{2,0} \ar[dl] \ar[dr] \\  &  & 𝕆_{0,4} \ar[dl] \ar[dr] &  & 𝕆_{2,2} \ar[dl] \ar[dr] &  & 𝕆_{4,0} \ar[dl] \ar[dr] \\  & 𝕆_{0,6} \ar[dl] \ar[dr] &  & 𝕆_{2,4} \ar[dl] \ar[dr] &  & 𝕆_{4,2} \ar[dl] \ar[dr] &  & 𝕆_{6,0} \ar[dl] \ar[dr] \\ 𝕆_{0,7} &  & 𝕆_{1,6} ∪ 𝕆_{2,5} &  & 𝕆_{3,4} ∪ 𝕆_{4,3} &  & 𝕆_{5,2} ∪ 𝕆_{6,1} &  & 𝕆_{7,0}}