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Here is a different one, on page 299: Proposition 17.3.13(iii) giving the relative root system for quasi-simple groups of type $D_{n}$ is not correct as stated. I believe the following is correct, but please comment if not!

If $n > rd$, then the relative root system has type $B_r$ if $d = 1$ and type $BC_r$ if $d > 1$. If $n = rd$, then the relative root system has type $D_r$ if $d = 1$ and type $C_r$ if $d > 1$. The case $n = rd$ only occurs when the group is an inner form (type ${}^1 D_{n,r}^{(d)}$ in Tits' notation).

(I don't know a reference. I didn't read this section of Springer, but I worked out the relative root datum from the absolute one in each case. Over a non-archimedean local field the answer agrees with Tits' Corvallis table.)

Updated: also part (i) is incorrect if $rd = n-1$. In this case the group is an outer form and the last element $a_{rd}$ in Springer's list should be replaced by the conjugate pair $a_{rd}$, $a_{rd+1}$ (i.e. $a_{n-1}$, $a_n$). See Tits' Boulder article from 1965.

Here is a different one, on page 299: Proposition 17.3.13(iii) giving the relative root system for quasi-simple groups of type $D_{n}$ is not correct as stated. I believe the following is correct, but please comment if not!

If $n > rd$, then the relative root system has type $B_r$ if $d = 1$ and type $BC_r$ if $d > 1$. If $n = rd$, then the relative root system has type $D_r$ if $d = 1$ and type $C_r$ if $d > 1$. The case $n = rd$ only occurs when the group is an inner form (type ${}^1 D_{n,r}^{(d)}$ in Tits' notation).

(I don't know a reference. I didn't read this section of Springer, but I worked out the relative root datum from the absolute one in each case. Over a non-archimedean local field the answer agrees with Tits' Corvallis table.)

Here is a different one, on page 299: Proposition 17.3.13(iii) giving the relative root system for quasi-simple groups of type $D_{n}$ is not correct as stated. I believe the following is correct, but please comment if not!

If $n > rd$, then the relative root system has type $B_r$ if $d = 1$ and type $BC_r$ if $d > 1$. If $n = rd$, then the relative root system has type $D_r$ if $d = 1$ and type $C_r$ if $d > 1$. The case $n = rd$ only occurs when the group is an inner form (type ${}^1 D_{n,r}^{(d)}$ in Tits' notation).

(I don't know a reference. I didn't read this section of Springer, but I worked out the relative root datum from the absolute one in each case. Over a non-archimedean local field the answer agrees with Tits' Corvallis table.)

Updated: also part (i) is incorrect if $rd = n-1$. In this case the group is an outer form and the last element $a_{rd}$ in Springer's list should be replaced by the conjugate pair $a_{rd}$, $a_{rd+1}$ (i.e. $a_{n-1}$, $a_n$). See Tits' Boulder article from 1965.

Source Link
fherzig
  • 1.4k
  • 1
  • 12
  • 13

Here is a different one, on page 299: Proposition 17.3.13(iii) giving the relative root system for quasi-simple groups of type $D_{n}$ is not correct as stated. I believe the following is correct, but please comment if not!

If $n > rd$, then the relative root system has type $B_r$ if $d = 1$ and type $BC_r$ if $d > 1$. If $n = rd$, then the relative root system has type $D_r$ if $d = 1$ and type $C_r$ if $d > 1$. The case $n = rd$ only occurs when the group is an inner form (type ${}^1 D_{n,r}^{(d)}$ in Tits' notation).

(I don't know a reference. I didn't read this section of Springer, but I worked out the relative root datum from the absolute one in each case. Over a non-archimedean local field the answer agrees with Tits' Corvallis table.)