Sports specific training

In my opinion the muscles which require strength and stability for skiing are:

Ankle Flexors.
If your boot is fitted correctly then any flex through the ankle will transfer directly through the boot into the ski. Ankle control and increased range of flexion into the ski edge will give smoother, faster and easier turns. Exercises: Edge Roll, Carving Stretch, Floor and Wall Stretch.

Inner leg from ankle to groin.
Increases the ability to hold your edge in difficult terrain and maintain ski width for parallel skiing. Exercises: Power Arc, Side Carve, Hip Roll.

Outer leg from ankle to hip.
Assists in controlling and supporting the knee and hip joint. These muscles stabilize the leg for smooth weight transfer over the skis. Strength and balance here gives you more ankle flexion and more edge in turns. Exercises: Edge Press, Side Lunge, Scissors.

Upper front leg. (Quadriceps)
This strength is vital in long runs, steep (black and above) grades, and bumps. Quad' strength helps you “lock” your posture over the boot; to lean forward into the front of the boot; so you spend less time in the back seat and at the edge of your balance. You need to ski inside your balance range all the time. Exercises: Side Lunge, Mogul Press, Edge Press.

Upper rear leg. (Glutes' and Hamstring)
When you're in the front of your boot and holding your line from edge to edge; the upper rear leg gives you thrust in the turn. They stabilize the hips and help keep them forward to avoid sitting “down' into the heels. Sitting down lifts the ski tips and immediately changes your balance point. Exercises: Side Lunge, Mogul Press, Edge Press.

Abdominals and lower back. These muscles create and maintain your core stability. Core stability is essential for pelvic balance. It helps you lean out over the mountain with control. It stabilizes your lower back to enable the use of your side abdominals which maintain an upright spine. Exercises: Scissors, Side Crunch, Wall Stretch.

Side Abdominals and side back.
Fantastic for getting extra pelvic lift without changing the line of your torso. They help you control your skis by taking pressure out of your hips through bumps and fast downhill turns. They help maintain your line down the mountain as your pelvis and legs move from side to side under that line. Exercises: Side Crunch, Carving Stretch, Wall Stretch.

Back (Lats' and Traps')
Keeps your spine upright and shoulders back for greater balance and arm extension, and better pole plants for turns. Exercises: Mogul Press, Side Carve, Power Arc, 1 Arm Row.

Rear Shoulders.
Supports the rotation of the arm at the shoulder, and helps your turn by giving greater strength to hold and pull onto the stock through the turn. Exercises: Power Arc, 1 Arm Row.

Hip Flexors.
Strong and supple hip flexors that respond quickly will minimize knee pressure through extreme angulation. They help take the strain of the turn away from the knee. Exercises: Hip Roll

BALANCE AND CORE STABILITY
Having a sense of balance on your skis allows you to feel into the ski. Feel ankle flexion, inner thigh strength, side abdominal contraction and posture stability. All are combined in split seconds to allow freedom on your skis.
Balance keeps you in the front of your boot over your ski and facing down the mountain at all times. Balance is first and last in having confidence on the slopes. Exercises: Edge Press, Side Lunge, Carving Stretch, Wall Stretch, Side Crunch, Scissors, Hip Roll.

STRETCH
Keeping your muscles supple, long and powerful, will give you the ability to correct situations quickly without overcompensating. Variable conditions on a slope; for example light powder over ice; where your edge sticks and then slides; require that your muscles be able to adapt quickly to the change under your ski so you keep your line and balance. Stretching regularly for muscles flexibility gives your muscle better tone to compensate as required without overstretching, which can result in injury. Exercises: Floor Stretch, Wall Stretch, Carving Stretch, Hip Roll.