KEY TO THE MAASS-SELBERG RELATIONS FOR SL_{3}(R)

λ_{1} = (it_{1} + ε_{1}, it_{2} + ε_{2}, it_{3} + ε_{3})
λ_{2} = (−it_{1}, −it_{2}, −it_{3})
s_{1}, s_{2} permutations in S_{3}

Inv(s) = {(i, j) | i < j, s(i) > s(j)}

α_{1}^{∨} = (1, −1, 0)
α_{2}^{∨} = (0, 1, −1)

Sample Box: s_{2}

s_{1} \begin{aligned}s_{1}λ_{1} + s_{2}λ_{2} & Inv(s_{2}) \\ & (s_{1}λ_{1} + s_{2}λ_{2})(α_{1}^{∨}) \\ Inv(s_{1}) \\ & (s_{1}λ_{1} + s_{2}λ_{2})(α_{2}^{∨}) \end{aligned}

\begin{matrix}eλ_{1} = (it_{1} + ε_{1}, it_{2} + ε_{2}, it_{3} + ε_{3}) & eλ_{2} = (−it_{1}, −it_{2}, −it_{3}) \\ (12)λ_{1} = (it_{2} + ε_{2}, it_{1} + ε_{1}, it_{3} + ε_{3}) & (12)λ_{2} = (−it_{2}, −it_{1}, −it_{3}) \\ (13)λ_{1} = (it_{3} + ε_{3}, it_{2} + ε_{2}, it_{1} + ε_{1}) & (13)λ_{2} = (−it_{3}, −it_{2}, −it_{1}) \\ (23)λ_{1} = (it_{1} + ε_{1}, it_{3} + ε_{3}, it_{2} + ε_{2}) & (23)λ_{2} = (−it_{1}, −it_{3}, −it_{2}) \\ (123)λ_{1} = (it_{3} + ε_{3}, it_{1} + ε_{1}, it_{2} + ε_{2}) & (123)λ_{2} = (−it_{3}, −it_{1}, −it_{2}) \\ (321)λ = (it_{2} + ε_{2}, it_{3} + ε_{3}, it_{1} + ε_{1}) & (321)λ_{2} = (−it_{2}, −it_{3}, −it_{1}) \end{matrix}

Inner Product Formula.

∫_{Γ\G/K} Λ^{c}E(g, λ_{1})Λ^{c}E(g, λ_{2})dg

= const. ∑_{s_{1},s_{2}∈S_{3}} \frac{e^{(s_{1}λ_{1}+s_{2}λ_{2})(c,0,−c)}}{(s_{1}λ_{1} + s_{2}λ_{2})(α_{1}^{∨})(s_{1}λ_{1} + s_{2}λ_{2})(α_{2}^{∨})}×

∏_{(j,k)∈ Inv s_{1}} \frac{Z(it_{j} − it_{k} + ε_{j} − ε_{k})}{Z(1 + it_{j} − it_{k} + ε_{j} − ε_{k})} ∏_{(j,k)∈ Inv s_{2}} \frac{Z(it_{k} − it_{j})}{Z(1 + it_{k} − it_{j})},

Z(s) = \sqrt{π} \frac{Γ(\frac{s}{2})ζ(s)}{Γ(\frac{s+1}{2})ζ(s + 1)}

Typeset by AMS-TEX

These boxes describe the ingredients of the 36 terms

s_{2} = e s_{2} = (12)

s_{1} = e \begin{matrix}[ \begin{matrix}ε_{1} \\ ε_{2} \\ ε_{3} \end{matrix} ] & ∅ \\ ∅ & {ε_{1} − ε_{2} \atop ε_{2} − ε_{3}} \end{matrix} \begin{matrix}[ \begin{matrix}it_{1} − it_{2} + ε_{1} \\ it_{2} − it_{1} + ε_{2} \\ ε_{3} \end{matrix} ] & 12 \\ ∅ & {2it_{1} − 2it_{2} + ε_{1} − ε_{2} \atop it_{2} − it_{1} + ε_{2} − ε_{3}} \end{matrix}

s_{1} = (12) \begin{matrix}[ \begin{matrix}it_{2} − it_{1} + ε_{2} \\ it_{1} − it_{2} + ε_{1} \\ ε_{3} \end{matrix} ] & ∅ \\ 12 & {2it_{2} − 2it_{1} + ε_{2} − ε_{1} \atop it_{1} − it_{2} + ε_{1} − ε_{3}} \end{matrix} \begin{matrix}[ \begin{matrix}ε_{2} \\ ε_{1} \\ ε_{3} \end{matrix} ] & 12 \\ 12 & {ε_{2} − ε_{1} \atop ε_{1} − ε_{3}} \end{matrix}

s_{1} = (13) \begin{matrix}[ \begin{matrix}it_{3} − it_{1} + ε_{3} \\ ε_{2} \\ it_{1} − it_{3} + ε_{1} \end{matrix} ] & ∅ \\ 23^{12 13} & {it_{3}−it_{1}+ε_{3}−ε_{2} \atop it_{3}−it_{1}+ε_{2}−ε_{1}} \end{matrix}  \begin{matrix}[ \begin{matrix}it_{3} − it_{2} + ε_{3} \\ it_{2} − it_{1} + ε_{2} \\ it_{1} − it_{3} + ε_{1} \end{matrix} ] & 12 \\ 12 13 & it_{1}+it_{3}−2it_{2}+ε_{3}−ε_{2} \\ 23 & it_{2}+it_{3}−2it_{1}+ε_{2}−ε_{1} \end{matrix}

s_{1} = (23) \begin{matrix}[ \begin{matrix}ε_{1} \\ it_{3} − it_{2} + ε_{3} \\ it_{2} − it_{3} + ε_{2} \end{matrix} ] & ∅ \\ 23 & {it_{2}−it_{3}+ε_{1}−ε_{3} \atop 2it_{3}−2it_{2}+ε_{3}−ε_{2}} \end{matrix}  \begin{matrix}[ \begin{matrix}it_{1} − it_{2} + ε_{1} \\ it_{3} − it_{1} + ε_{3} \\ it_{2} − it_{3} + ε_{2} \end{matrix} ] & 12 \\ 23 & {2it_{1}−it_{2}−it_{3}+ε_{1}−ε_{3} \atop 2it_{3}−it_{1}−it_{2}+ε_{3}−ε_{2}} \end{matrix}

s_{2} = e s_{2} = (12)

s_{1} = (123) \begin{matrix}[ \begin{matrix}it_{3} − it_{1} + ε_{3} \\ it_{1} − it_{2} + ε_{1} \\ it_{2} − it_{3} + ε_{2} \end{matrix} ] & ∅ \\ 13 & it_{2}+it_{3}−2it_{1}+ε_{3}−ε_{1} \\ 23 & it_{1}+it_{3}−2it_{2}+ε_{1}−ε_{2} \end{matrix}  \begin{matrix}[ \begin{matrix}it_{3} − it_{2} + ε_{3} \\ ε_{1} \\ it_{2} − it_{3} + ε_{2} \end{matrix} ] & 12 \\ 13 & it_{3}−it_{2}+ε_{3}−ε_{1} \\ 23 & it_{3}−it_{2}+ε_{1}−ε_{2} \end{matrix}

s_{1} = (321) \begin{matrix}[ \begin{matrix}it_{2} − it_{1} + ε_{2} \\ it_{3} − it_{2} + ε_{3} \\ it_{1} − it_{3} + ε_{1} \end{matrix} ] & ∅ \\ 12 & 2it_{2}−it_{1}−it_{3}+ε_{2}−ε_{3} \\ 13 & 2it_{3}−it_{1}−it_{2}+ε_{3}−ε_{1} \end{matrix}  \begin{matrix}[ \begin{matrix}ε_{2} \\ it_{3} − it_{1} + ε_{3} \\ it_{1} − it_{3} + ε_{1} \end{matrix} ] & 12 \\ 12 & it_{1}−it_{3}+ε_{2}−ε_{3} \\ 13 & 2it_{3}−2it_{1}+ε_{3}−ε_{1} \end{matrix}

s_{2} = (13) s_{2} = (23)

s_{1} = e \begin{matrix}[ \begin{matrix}it_{1} − it_{3} + ε_{1} \\ ε_{2} \\ it_{3} − it_{1} + ε_{3} \end{matrix} ] & 12 13 23 \\ ∅ & {it_{1}−it_{3}+ε_{1}−ε_{2} \atop it_{1}−it_{3}+ε_{2}−ε_{3}} \end{matrix}  \begin{matrix}[ \begin{matrix}ε_{1} \\ it_{2} − it_{3} + ε_{2} \\ it_{3} − it_{2} + ε_{3} \end{matrix} ] & 23 \\ ∅ & {it_{3}−it_{2}+ε_{1}−ε_{2} \atop 2it_{2}−2it_{3}+ε_{2}−ε_{3}} \end{matrix}

s_{1} = (12) \begin{aligned}[ \begin{matrix}it_{2} − it_{3} + ε_{2} \\ it_{1} − it_{2} + ε_{1} \\ it_{3} − it_{1} + ε_{3} \end{matrix} ] & 12 13 23 \\ 12 & 2it_{2}−it_{1}−it_{3}+ε_{2}−ε_{1} \\ & 2it_{1}−it_{2}−it_{3}+ε_{1}−ε_{3} \end{aligned}  \begin{aligned}[ \begin{matrix}it_{2} − it_{1} + ε_{2} \\ it_{1} − it_{3} + ε_{1} \\ it_{3} − it_{2} + ε_{3} \end{matrix} ] & 23 \\ 12 & it_{2}+it_{3}−2it_{1}+ε_{2}−ε_{1} \\ & it_{1}+it_{2}−2it_{3}+ε_{1}−ε_{3} \end{aligned}

s_{1} = (13) \begin{matrix}[ \begin{matrix}ε_{3} \\ ε_{2} \\ ε_{1} \end{matrix} ] & 12 13 23 \\ 12 13 23 & {ε_{3} − ε_{2} \atop ε_{2} − ε_{1}} \end{matrix} \begin{matrix}[ \begin{matrix}it_{3} − it_{1} + ε_{3} \\ it_{2} − it_{3} + ε_{2} \\ it_{1} − it_{2} + ε_{1} \end{matrix} ] & 23 \\ 12 13 & 2it_{3}−it_{1}−it_{2}+ε_{3}−ε_{2} \\ 23 & 2it_{2}−it_{3}−it_{1}+ε_{2}−ε_{1} \end{matrix}

s_{1} = (23) \begin{aligned}[ \begin{matrix}it_{1} − it_{3} + ε_{1} \\ it_{3} − it_{2} + ε_{3} \\ it_{2} − it_{1} + ε_{2} \end{matrix} ] & {12 13 \atop 23} \\ 23 & it_{2}+it_{1}−2it_{3}+ε_{1}−ε_{3} \\ & it_{1}+it_{3}−2it_{2}+ε_{3}−ε_{2} \end{aligned}  \begin{aligned}[ \begin{matrix}ε_{1} \\ ε_{3} \\ ε_{2} \end{matrix} ] & 23 \\ 23 & ε_{1} − ε_{3} \\ & ε_{3} − ε_{2} \end{aligned}

s_{2} = (13) s_{2} = (23)

s_{1} = (123) \begin{matrix}[ \begin{matrix}ε_{3} \\ it_{1} − it_{2} + ε_{1} \\ it_{2} − it_{1} + ε_{2} \end{matrix} ] & 12 13 23 \\ 13 & it_{2}−it_{1}+ε_{3}−ε_{1} \\ 23 & 2it_{1}−2it_{2}+ε_{1}−ε_{2} \end{matrix}  \begin{aligned}[ \begin{matrix}it_{3} − it_{1} + ε_{3} \\ it_{1} − it_{3} + ε_{1} \\ ε_{2} \end{matrix} ] & 23 \\ 13 23 & 2it_{3}−2it_{1}+ε_{3}−ε_{1} \\ & it_{1}−it_{3}+ε_{1}−ε_{2} \end{aligned}

s_{1} = (321) \begin{matrix}[ \begin{matrix}it_{2} − it_{3} + ε_{2} \\ it_{3} − it_{2} + ε_{3} \\ ε_{1} \end{matrix} ] & 12 13 23 \\ 12 & 2it_{2}−2it_{3}+ε_{2}−ε_{3} \\ 13 & it_{3}−it_{2}+ε_{3}−ε_{1} \end{matrix}  \begin{aligned}[ \begin{matrix}it_{2} − it_{1} + ε_{2} \\ ε_{3} \\ it_{1} − it_{2} + ε_{1} \end{matrix} ] & 23 \\ 12 13 & it_{2}−it_{1}+ε_{2}−ε_{3} \\ & it_{2}−it_{1}+ε_{3}−ε_{1} \end{aligned}

s_{2} = (123) s_{2} = (321)

s_{1} = e \begin{aligned}[ \begin{matrix}it_{1} − it_{3} + ε_{1} \\ it_{2} − it_{1} + ε_{2} \\ it_{3} − it_{2} + ε_{3} \end{matrix} ] & 13 23 \\ ∅ & 2it_{1}−it_{2}−it_{3}+ε_{1}−ε_{2} \\ & 2it_{2}−it_{1}−it_{3}+ε_{2}−ε_{3} \end{aligned}  \begin{aligned}[ \begin{matrix}it_{1} − it_{2} + ε_{1} \\ it_{2} − it_{3} + ε_{2} \\ it_{3} − it_{1} + ε_{3} \end{matrix} ] & 12 13 \\ ∅ & it_{3}+it_{1}−2it_{2}+ε_{1}−ε_{2} \\ & it_{1}+it_{2}−2it_{3}+ε_{2}−ε_{3} \end{aligned}

s_{1} = (12) \begin{aligned}[ \begin{matrix}it_{2} − it_{3} + ε_{2} \\ ε_{1} \\ it_{3} − it_{2} + ε_{3} \end{matrix} ] & 13 23 \\ 12 & it_{2}−it_{3}+ε_{2}−ε_{1} \\ & it_{2}−it_{3}+ε_{1}−ε_{3} \end{aligned}  \begin{aligned}[ \begin{matrix}ε_{2} \\ it_{1} − it_{3} + ε_{1} \\ it_{3} − it_{1} + ε_{3} \end{matrix} ] & 12 13 \\ 12 & it_{3}−it_{1}+ε_{2}−ε_{1} \\ & 2it_{1}−2it_{3}+ε_{1}−ε_{3} \end{aligned}

s_{1} = (13) \begin{matrix}[ \begin{matrix}ε_{3} \\ it_{2} − it_{1} + ε_{2} \\ it_{1} − it_{2} + ε_{1} \end{matrix} ] & 13 23 \\ 12 13 & it_{1}−it_{2}+ε_{3}−ε_{2} \\ 23 & 2it_{2}−2it_{1}+ε_{2}−ε_{1} \end{matrix}  \begin{matrix}[ \begin{matrix}it_{3} − it_{2} + ε_{3} \\ it_{2} − it_{3} + ε_{2} \\ ε_{1} \end{matrix} ] & 12 13 \\ 12 13 & 2it_{3}−2it_{2}+ε_{3}−ε_{2} \\ 23 & it_{2}−it_{3}+ε_{2}−ε_{1} \end{matrix}

s_{1} = (23) \begin{aligned}[ \begin{matrix}it_{1} − it_{3} + ε_{1} \\ it_{3} − it_{1} + ε_{3} \\ ε_{2} \end{matrix} ] & 13 23 \\ 23 & 2it_{1}−2it_{3}+ε_{1}−ε_{3} \\ & it_{3}−it_{1}+ε_{3}−ε_{2} \end{aligned}  \begin{aligned}[ \begin{matrix}it_{1} − it_{2} + ε_{1} \\ ε_{3} \\ it_{2} − it_{1} + ε_{2} \end{matrix} ] & 12 13 \\ 23 & it_{1}−it_{2}+ε_{1}−ε_{3} \\ & it_{1}−it_{2}+ε_{3}−ε_{2} \end{aligned}

s_{2} = (123) s_{2} = (321)

s_{1} = (123) \begin{aligned}[ \begin{matrix}ε_{3} \\ ε_{1} \\ ε_{2} \end{matrix} ] & 13 23 \\ 13 23 & ε_{3} − ε_{1} \\ & ε_{1} − ε_{2} \end{aligned} \begin{aligned}[ \begin{matrix}it_{3} − it_{2} + ε_{3} \\ it_{1} − it_{3} + ε_{1} \\ it_{2} − it_{1} + ε_{2} \end{matrix} ] & 12 13 \\ 13 23 & 2it_{3}−it_{1}−it_{2}+ε_{3}−ε_{1} \\ & 2it_{1}−it_{3}−it_{2}+ε_{1}−ε_{2} \end{aligned}

s_{1} = (321) \begin{aligned}[ \begin{matrix}it_{2} − it_{3} + ε_{2} \\ it_{3} − it_{1} + ε_{3} \\ ε_{1} \end{matrix} ] & 13 23 \\ 12 13 & it_{1}+it_{2}−2it_{3}+ε_{2}−ε_{3} \\ & it_{3}−it_{1}+ε_{3}−ε_{1} \end{aligned}  \begin{aligned}[ \begin{matrix}ε_{2} \\ ε_{3} \\ ε_{1} \end{matrix} ] & 12 13 \\ 12 13 & ε_{2} − ε_{3} \\ & ε_{3} − ε_{1} \end{aligned}